The anecdote popped up in a piece Rachel Johnson wrote for the New Statesman last year, where she gleefully concluded that, like rats, "in London you're never more than a few feet from at least two Johnsons". She was only exaggerating slightly.

"No! Not another one! How many more of you Johnsons are there?" David Cameron is said to have exclaimed when he met Julia, Boris’s half-sister. Julia, we’re told, then took great pleasure in introducing the then-Prime Minister to Max, her other brother.

This piece of advice neatly sums up the Johnson ethos: unwavering belief of one's intellectual abilities, a staunch refusal to take anything seriously and the compulsion to never say anything privately if it can instead be performed in front of an audience.

"You may have to wing it from time to time," he said, "but if you can play Richard II in the [Eton] Cloisters without knowing the part, you’ll probably get away with it."

When Boris Johnson announced his intentions to run for London mayor in 2007, his father Stanley, a former MEP, environmentalist and World Bank employee, wrote a column in the Spectator. In it, he hit out against those already calling his son a "buffoon", explaining that though he wasn’t always organised (he once played the title role in Richard II while at Eton and made up his own lines as he'd not learnt Shakespeare's), he could very well do the job.

What do they really want, and can anything stop them from getting it?

From the Foreign Office to Jo Johnson’s post at the Department for Transport, Stanley Johnson’s appearance on I’m A Celebrity Get Me Out Of Here and Rachel Johnson’s turn on Celebrity Big Brother, via endless columns in the Mail on Sunday and Daily Telegraph, the family have set out to conquer Britain.

Boris Johnson, the Foreign Secretary and two-term London Mayor, was the first of the brood to make his mark on public life, but it now feels impossible to ignore the relentless rise of the boisterous, irritating, enviable clan that’s taken over politics, the media and everything in between.

According to Charlotte, the toddler looked on with "shock, disbelief and fear" when the Johnson clan welcomed its latest arrival, and competition soon began between the two.

The young Boris led a charmed existence typical of the first child born into a bright, young, upper-middle class family, but his life was soon shattered with the arrival of his little sister Rachel, just over a year after he was born.

"Stanley lit the touch paper and asked, 'Which of you is going to get to the very top?'"

Stanley married the painter Charlotte Johnson-Wahl in 1963. When she gave birth to her first child, Boris, a year later, Stanley wasn’t present: he’d popped out of the hospital to buy himself a slice of pizza.

Some years later, he attempted to get selected as a Conservative parliamentary candidate in Leicester, and proudly told the panel that, until then, the nearest he’d ever been to his prospective constituency was Leicester Square. He did not get selected.

Stanley's early years give a sense of where the Johnson spawn inherited their audacity. When he was studying at Oxford he discovered the Newdigate, a "prize for a long original poem on some arcane subject", according to a profile of him published in the Daily Mail in 2004. "Stanley found out about it only about midnight before the 9AM deadline. He promptly sat down and knocked out 100 or so deftly rhyming couplets before breakfast and won hands-down."

"It’s the worst part of the English establishment, people who are brought up to be facile, clever, funny – it’s the sort of 'everything is a game, I’m clever, I’m witty, I’m funny enough to disarm you, but effectively it’s just a sword fight, and if I succeed and you fail then that’s great, we’ll have a laugh' attitude," explained a source who knows some of the Johnson siblings.

By 1999, he was made editor of the Spectator, a role he could only secure by promising that he wouldn’t stand as an MP. Around the same time he had also promised his local Conservative association that he would leave the Spectator if he were to join the House of Commons. Naturally, he was elected in Henley in 2001 and kept his editorship.

The rest, in this case, is very much history: first there was Eton, where Boris became secretary of the debating society and editor of the school newspaper, then Oxford where he joined the Bullingdon Club and was elected president of the union. In 1989, he became a trainee at the Times and got fired after fabricating quotes, moved to the Daily Telegraph having met the editor while at university, moved to Brussels and became one of the most passionately eurosceptic journalists there, came back, and filed columns to whoever would have them.

Sonia Purnell, who wrote Just Boris: The Irresistible Rise of a Political Celebrity, agreed. "It’s interesting to see how families behave, and with the Johnsons, there was a feeling that a few generations ago they were terribly grand, and their rightful place in society has been taken away from them, so they were all going to have to compete to get back at the top of the pile, the biggest and blondest dynasty of all," she said. "Stanley lit the touch paper and asked, 'Okay, which of you is going to get to the very top?' and Boris was the eldest sibling, so it was always going to be him."

"Stanley deliberately created a family atmosphere in which beating the others at running, jumping, eating the hottest mince pies, coming first at school or simply having the blondest hair entirely captured the lives of all four children from his first marriage," remarked a profile of the family in the Sunday Times in 2013.

On this topic, like others, the anecdotes are seemingly endless: Boris once broke his toe by kicking a table after losing a point in a table tennis game to Rachel; Leo once landed Boris in hospital by shooting him in the stomach with an air gun when he was meant to aim at tin cans, and so on.

The sister learnt how to read before her brother did – "my paternal grandmother used to ask me to read out Times leaders when I was four", she explained once – but with the later births of Leo and Jo, the house became the scene of a never-ending war.

"Boris was a poor manager, administrator and politician, apart from the campaigning, which he’s brilliant at, and the same thing is happening at the Foreign Office. What you hear from officials there is that there's no direction, there are no ideas, it’s all chaos and he doesn’t care, but nor does he seem competent to do anything about it."

One thing that has remained consistent is his propensity for blunders: "The interesting thing about Boris is that competence is a real issue," Purnell said. "When he first became mayor of London, it was utter chaos. The chief of staff in City Hall had to take him out to dinner and say 'For god’s sake, get a grip, start behaving like a mayor, do your job.'

Fifteen years later, Boris's political career has encompassed a series of extramarital scandals, two successful London mayoral runs, a surprise campaign for Brexit and two messy years at the Foreign Office – yet these questions still haven't been answered. If anything, Boris’s beliefs and motivations are less clear now than they were when he was writing leaders in the Spectator every week.

Already a rising star in the party two years later, Boris was interviewed by Lynn Barber for the Guardian, and she wrote of her encounter: "Johnson is so larky, so ready to retract an opinion, or agree with criticism, that it is very difficult to sort out what he really believes. What are his sticking points? What principles would he go to the barricades for? What would he consider a resigning matter?"

She was also one of the only three girls at her boarding school, which she wrote about in the Telegraph in 2011, cheerfully explaining: "It was at Ashdown that I discovered boys. It wasn’t hard. I was absolutely submerged in them. I would choose one – like the King of Samarkand in the tale of Scheherazade – and send him a note: 'Meet me by the ha-ha at 6PM.' They would trot along, and then we would kiss."

Someone else shares that trait with him: his younger sister, Rachel – which perhaps isn’t surprising, given she got to experience Boris’s fierce competitiveness from the earliest age. A journalist by trade, Rachel was the Financial Times’ first ever female graduate trainee, before moving on to the BBC, becoming a columnist and editing society magazine The Lady, all while publishing a series of well-selling novels.

"What I found amazing was hearing someone who very well may be Prime Minister one day, and he was worried that I was angry with him," he said, "and this is a breathtaking weakness in a politician – he wants to be loved even by the people he’s destroying."

Boris's all-encompassing ambition sits side-by-side with his yearning for attention, and his near-pathological desire to be liked. In the 2013 documentary Boris Johnson: The Irresistible Rise, Ken Livingstone was interviewed about his mayoral rival and spoke candidly of the way Boris had come to talk to him after one of their many slanging matches on the radio during the 2012 campaign.

Still, he ploughs on, and those who know him think his desire to become Prime Minister is as present as ever – but why does he do it? "He’s hard-wired to win, whether it’s ping-pong or an election, he cannot bear the thought of losing," Purnell explained.

"On Question Time on Thursday, the Johnson mouthpiece appeared reluctant to answer any question that was not about big bro," an Independent piece from 2008 dryly noted. "On his election, she promised 'many years of Boris-induced sunshine' in London. On the Tory party, she announced that he had 'brought them to a collective orgasm'. Asked about Scottish devolution, she stammered: 'I'm not sure I will be physically able to answer a question that's not about my brother.'"

Career aside, Rachel has another job which takes up nearly as much of her time: cheerleading for her family, a position she’s been occupying for decades.

She joined Celebrity Big Brother in January, was the second guest to be evicted, but mostly used it as an excuse to write multiple columns about it beforehand and afterwards.

Indeed, looking only at the past few months shows how much of a rollercoaster Rachel's life can be: she grabbed the headlines by becoming a Liberal Democrat before the election as a protest against Brexit, then stopped talking about her new party altogether.

"She’s a gem, she’s mad – she’s one of those madly intelligent people who gets bored very quickly, so tends to not always give you her full attention," said another who used to work for her at The Lady. "She tends to move on very quickly, but I think she just wants to have a lot of fun all the time, and she does it without malice."

"Things are a game for her; she’s fun and doesn’t take herself too seriously, and she’s smart, but she’s not a deep, committed, meaningful person," said one journalist who shared a newsroom with her.

Her glee doesn’t stop at getting to pick and choose from a sea of teenage boys; talking to people who know and have worked with her, being Rachel Johnson means being lightly unburdened by the realities of life, and having a grand old time as a result.

"The last thing people want to do is hear about another bloody Johnson. We’ve got a glut of Johnsons. My dream is to have some sort of Google counter-alert that removes any newspaper article referring to a Johnson," he said in one interview .

Rachel speaks fondly of the other middle-child in the Johnson clan, Leo. In 2015, she told the Guardian, "When he was little he’d jump on to the table and dance reggae-style. He’s in a band, not that he’d tell you. He doesn’t seek the spotlight like the rest of us." Leo is also the only sibling not to have the instantly recognisable bright blond hair, and the only one who has never had a job in politics or newspapers. His career has mostly focused on issues of environment and sustainability.

Jo’s political career was never the obvious choice: after studying in England and Brussels – and a short stint as an investment banker – he joined the Financial Times and stayed there for over a decade. His path at the pink paper led him to Paris, where he was a correspondent, then New Delhi, where he became South Asia bureau chief, before returning to London and being made associate editor and head of the paper's influential Lex column.

Speaking about Jo, the youngest of the four, he said in 2013: "He’s the tallest, he’s the blondest, he’s the cleverest. But, if I’m being honest with you, I felt he broke ranks. He and I had this deal, though it was never quite explicit, that we would live these lives of peace, normally, doing half-interesting things that we care about. And then he announced he was going to stand to become a Tory MP."

Still, the shy brother would never pass an opportunity to talk up his clan and toy with his siblings' ambition and need for constant comparison and competition in the process.

Leo also tries not to make his relationship with his family too symbiotic, explaining that his wife and him had agreed to frequently have some JFWs – Johnson-free weekends – which he uses as leverage when he needs something from her.

"I’m the non-political one. I’m not blond. I’m not Tory. I’m born with the gene for self-publicity missing, or at least defective. It comes on and off, and when it comes on, no one is interested," he said in an interview with the Evening Standard.

Does this make him the boring Johnson? Not necessarily. Someone who worked with him during his time at the department for education said: "There’s a big contrast – you watch him in debates and he seems quite dry, a bit boring, his voice is a bit monotone. Then, in one-on-one interactions and in meetings, he's actually really friendly and also really funny."

He has remained true to form since then, becoming the head of Number 10's policy unit, then Minister for Universities and Science, while, according to one political editor, "firmly and elegantly declining all efforts to get to know him".

Quoting what one of his Oxford contemporaries had told Private Eye that week – "he could not be more different to Boris, it’s as though the humour gene by-passed Jo altogether and he inherited only the ambition gene" – he added that he would "not really apologise for the humour-ectomy, nor, indeed, for any hint of ambition that the House might detect".

But he didn’t, and was elected as the Conservative MP for Orpington in 2010. His maiden speech to the House of Commons set the tone for what was to follow. "Anyone hoping that I will enliven proceedings in the manner of my elder brother, the former Member for Henley, is likely to be disappointed," he told the chamber.

"Jo is different; he’s more awkward and he does come across as a bit more shy. He doesn’t have the Johnson exuberance and show-offiness that the others have," they added, saying colleagues expected him to hang around.

A journalist who worked alongside him there said he was "like a classic FT reporter in lots of ways, serious reporter, serious thinker, working his way up".

Heaps of misogynistic and all-round bigoted tweets were unearthed from Young’s account, and he eventually stepped down from the post, but not before being publicly defended by both Boris and Jo.

Then, in a move that is rumoured to have contributed to Number 10’s decision to give him the not-so-glamorous job of Minister for Transport and London, Johnson the Youngest appointed Toby Young to the Office for Students.

A famous anecdote involves Rachel calling up Boris, who’d only got a 2:1 at his degree, after Jo graduated. "Have you heard the bad news about Jo?" she asked her brother. "He got a first."

The stories have repeatedly featured on the front pages of right-leaning newspapers over the past year, and the details change each time, but the template – "hysterical left-wing students have decided to ban/no platform seemingly offensive book/movie/guest speaker which proves that free speech is in danger" – remains the same.

He has, however, taken a bit of a sharp turn over the past few months. First, there was the war against so-called snowflakes and free speech denialists on campus.

The family’s trademark cockiness is also there. "He was really driven and he wasn’t hated that much," they added. "People felt he didn’t understand higher education, but they would really respect him because he engaged. He’d come to an awards dinner for Times Higher Education in a room full of academics who hated all the stuff he pushed through and would do a cracking speech with a lot of jokes about how they hated the reforms, and people would laugh at them."

As enjoyable as that prospect would be for the Johnson Sr, the very different brothers share one common obstacle: neither has a lot of political allies around them.

Unsurprisingly, the Johnson patriarch would dearly enjoy the idea of a Tory leadership contest turned Double-Johnson extravaganza: "Is that going to happen in some distant future?" Stanley pondered in a Mail on Sunday column in 2013. "Frankly, I haven't the faintest idea. But if it did, I am sure that – from a spectator point of view, at least – it would be tremendous fun."

A famous anecdote involves Rachel calling up Boris, who’d only got a 2:1 at his degree, after Jo graduated. "Have you heard the bad news about Jo?" she asked her brother. "He got a first."

While Westminster has been in turns charmed, bored and infuriated by Boris’s naked attempts to become Prime Minister, Jo has been making his way up quietly, and could yet pip him at the post. This would infuriate Boris: though he always competed against his closest sibling, Rachel, he was always jealous of the intellectual achievements of his baby brother.

Though uncharacteristic of a Remain-supporting politician who had so far erred towards the centre of his party, this might well mean that Jo has Downing Street ambitions of his own and is trying to appeal to the Conservatives he’s failed to reach so far.

"The free speech stuff never came up [when we worked together]. He gets no platforming and free speech, and he doesn’t think it’s an issue, so it looks like media positioning, trying to get some quick hits with the Telegraph or the Times," said an education sector source, surprised by Jo’s sudden adoption of a Boris-like antipathy to safe spaces. "The Toby Young thing, too – when I saw it I thought, 'Oh, he’s actually lost his shit.'"

Rachel also emphasises this when asked, telling t_he Sunday Times_ in 2013 that apart from Boris going to visit a friend called Carl once, "we never, ever, went to play with other children. We didn’t need friends."

Then, there was the influence of their mother: while the children were growing up, Charlotte’s mental health quickly deteriorated, and she describes in interviews how severe OCD rendered her barely capable of taking care of her family. This culminated in a months-long stay in a psychiatric hospital. "When I was in hospital, they became very close to each other because we had a series of dotty nannies and housekeepers looking after them, and they grew very close and very protective of each other," she explained in Boris Johnson: The Irresistible Rise.

First, there was the chaos of their childhood; because of Stanley’s messy career, the children moved around constantly – Boris had lived in over 30 different homes by the time he was 14.

This shouldn’t come as a surprise. As hard as it can be to avoid bumping into at least one Johnson when out in well-heeled establishments in the capital, the clan has always strived to be self-sufficient.

Much has been made of Boris's lack of a "caucus" within the parliamentary Conservative party – he doesn’t court other MPs, is rarely seen out on the terrace bar in Parliament – but he isn’t the only lonely one. Though broadly well-respected, Jo doesn’t have a gang around him either, and regardless of what happens next in the party, grabbing the leadership without a support base might prove impossible.

Though one can assume she was at least partly joking, people who know her well agree that the Johnsons do have a real sense of inadequacy compared to their posher peers. "It's that thing of 'we were brought up with very very posh people, but actually we’re not that posh, we don’t have a title or a big house', which is obviously ridiculous to everyone else who’s not that posh," one of them said. "I think one of the reasons why Boris Johnson wanted Foreign Secretary so much was for Chevening [the Grade I listed building in Kent that comes with the job]. Those guys are always there – Christmas, Easter, all the Johnsons go."

In that same interview, in fact, Rachel mentions her aristocrat husband’s hopefully tongue-in-cheek remarks about her pedigree: "His family have proved all they've got to prove, whereas we haven't proved anything. Ivo's descended from the poshest of the posh – he's unbelievably blue-blooded. He tells people he had to marry me to introduce some thick peasant stock into the Dawnay family gene pool."

The family is also descended from, among others, Turkey and Germany, which makes them stand out next to the families they brushed shoulders with by sending their children to the country’s top public schools.

Putting scaled-down violins aside, the family is a reminder that, for some, class can be relative, and the Johnsons have always floated around more established circles than theirs.

"The thing about the Johnsons is, we don't have any money. We have to earn our keep," Rachel told Tatler in 2008.

Beyond all this lies something else, without which no story of a British family could be complete: class.

This might well be the real reason behind the brood's fiery ambition and compulsive desire to be seen and heard and loved by all.

"The outsider will always have a fascination for getting in, particularly with British families who have that extended genealogy to them," explained Joy Lo Dico, the editor o_f the Evening Standard__’s_ Londoner’s Diary, which has long charted the Johnsons’ rise. "What’s happened by accident is that we have ended up with a family that itself has become a dynasty, rather than attaching itself to any particular line."

Nowhere is this more obvious than in the the press. Reading up on the Johnsons means falling into a seemingly never-ending spiral of Boris talking about Rachel, Rachel talking about Stanley, Leo talking about Jo, Jo talking about Boris, ad nauseam.

Their near-pathological obsession with getting a seat at the table has seen them taking over the table entirely, and as long as they keep supporting each other and acting like it’s the Johnsons against the world, few outsiders can stop them.

Boris might well have missed his shot at becoming Prime Minister, but it doesn’t matter any more: he merely is the brightest star in a furiously entitled and charmingly obnoxious constellation.

If their downfall must come, it will probably have to come from the inside. The eldest’s jealousy at his youngest brother’s success could fracture the family; Rachel and the others could decide that Boris’s shameless Brexit obsession finally is a step too far.

Or they could stick together, for better or worse, and the for the sake of their own dominance, and the generations of Johnsons yet to come.

The next one is already on the march: two years ago, Lara Johnson-Wheeler, Boris's 20-something daughter, penned a piece for the Evening Standard on – you guessed it – her family’s work.

"I swear this isn’t another piece of Johnson self-promo," she helpfully announced. "I swear this is in writing because I have been working at the exhibition all week and I have been asked a countless number of times why the retrospective exhibition currently on at Mall Galleries hasn’t had enough publicity."

The exhibition she mentioned was, of course, her grandmother’s.

Equally enamoured with the many Johnsons, the Telegraph then took to commissioning Oliver Dawnay, Rachel’s son, to write about his grandfather’s stint on I’m A Celebrity Get Me Out of Here.

Like a perpetual motion machine, the Johnsons never stop talking about one another, making each other more popular in the process – and like a bleached blond Hydra, it can’t be too far fetched to imagine three more messy yellow mops popping up if one disappears.

Or maybe – maybe – the children of Boris, Rachel, Jo and Leo will decide that, having grown up in the spotlight already, there is no point in them continuing their parents' crusades.