Grey clouds hang heavy in the skies as the scout troop marches into the cemetery, followed by a noisy brass band made up of scruffy teenagers in trainers and elderly men in suits. Behind them tramp hundreds of Reykjavik residents, many carrying Iceland's flag.

The music stops, then the crowd gathers round as a woman in traditional dress places a large wreath on a grave. It is Independence Day, and people are here to honour the 200th anniversary of the man whose birth this day marks: Jón Sigurdsson, hero of the nation's struggle for freedom from Danish control.

Overseeing it all is a burly man with a sharp haircut. People come up to greet him. Some take their photograph with him. He smiles, then chats to environmental campaigners carrying banners against political corruption, one of which features a picture of Jón Sigurdsson with a wordplay on his most famous slogan, changing "We all protest" to "We all puke".

This is Jón Gnarr, the unlikely mayor of Iceland's capital, who delivered a huge shock to the political system in the wake of the country's financial meltdown. One year ago the comedian led a gang of ex-punks, poets and pop stars to control of city hall. They call themselves anarcho-surrealists and their aim is to transform politics.

A stylish blonde woman in a bright green coat bounds up to say hello. Eva Einarsdóttir, a human rights activist nicknamed "Palestine Eva", is chairwoman of the national day committee and has arrived fresh from a ceremony with the president and prime minister. How was it? "Fine," she says. "I managed not to laugh, although it was hard at times."

Such irreverence has made the Best Party loved and loathed in equal measure. It burst on to the scene before last year's municipal election, satirising politics and throwing traditional parties into disarray.

Its first pledge was to break all its promises, making the party almost impossible to attack, then it promised a polar bear to the zoo and a drug-free parliament within 10 years.

The party's only advertisement was in a newspaper personal column, saying: "The Best Party wishes to meet good people aged between 18 and 90." Its 10-point plan had 13 points. And the party's campaign video featured candidates singing Tina Turner's Simply the Best, with the chorus: "We are the best, the bestest of parties, best for Reykjavik, best city of every week." It works in Icelandic.

Iceland was ripe for change, having effectively gone bust thanks to the cronyism of a cluster of politicians and bankers who thought that they could turn an island of fishermen with a population of 318,000 into a financial superpower.

In less than four years, the most rapid expansion of a banking system in history saw three privatised banks develop assets 10 times the size of the country's GDP. It was the Icarus economy. Property prices tripled, the stock market multiplied nine times, and people borrowed heavily – often in foreign currencies – to cash in on the boom. The crash was fast, hard and painful, worsened by the collapse of the krona as the state, unable to bail out the banks, refused to pay foreign creditors.

The strategy looks smart now, compared with events in Greece and Ireland, but the country was angry and frightened. Voters wanted change, and the Best Party caught the mood, capturing Reykjavik with 34.7% of the vote. "No one has to be afraid of the Best Party," said Jón in his acceptance speech. "Because it is the best party. If it wasn't, it would be called the Worst Party or the Bad Party. We would never work with a party like that."

The tone was jocular, but the intent deadly serious, as becomes clear talking to Jón and Einar Örn Benediktsson, who used to sing alongside Björk in the Sugarcubes and is now Reykjavik's head of culture and tourism. We dine on guillemot and lamb in a harbourside restaurant, the infamous whaling boats bobbing outside the window, while the pair explain their mission to transform political discourse.

"The human spirit has been crushed by small-minded people playing politics," says Einar Örn. "We have no agenda and are just fully engaged in trying to do our best. We have no party members and no idea about spin or political punchlines. When we don't know something, we admit ignorance."

Given the frivolity of their campaign and the fact the mayor is a famous comedian, best known for playing a bad-tempered Marxist on television, they have surprised people with the seriousness they have shown in running the city. They claim their campaign promises had serious undertones – the polar bear pledge, for example, was a satirical comment on attitudes to immigration and climate change, since four of the creatures have been shot in recent years after swimming into Icelandic waters.

They freely admit they have made mistakes, confronted with huge political problems in a city inhabited by nearly half the Icelandic people. They had to slash Reykjavik's spending by 10%, which involved cutting bus services and putting up charges for services, while Jón oversaw the sacking of 70 staff as he grappled with a mismanaged utility company with debts five times the size of the city budget.

Jón has found the role highly stressful, becoming ill as he struggled with complex issues. He left school at 16 after a troubled adolescence, during which he was a heavy glue sniffer and was sent to a special boarding school. Giving a mayoral speech at an award to a social worker, he burst into tears, wondering if he was cutting the sort of services that helped him as a teenager. He was relieved to find he wasn't.

He describes himself as an anarchist, inspired by Gandhi, Tolstoy, Bakunin and the British punk band, Crass. His radio shows have included crank calls to the CIA and the White House, but he claims to have known nothing of politics beyond what he had seen on The Wire. "I just love fucking with narrow-minded people who take themselves too seriously. I really like to irritate arrogant people, all those authoritarian people who want to control what we say and do."

He has banned religious groups from city schools, dressed in drag to mark gay pride, and had the city's sky-blue crest tattooed on his left forearm. After his summer holiday, he plans to paint his nails, wear lipstick and campaign for great apes to be given human rights. "I might try resigning on a really minor issue that I get wrong, since no one here has resigned for all the huge things that went wrong."

Such tactics infuriate the traditional parties, who deride them as jokers and amateurs. "Because of their lack of experience, the officials have gained power, which is not a good thing," said Kjartan Magnússon of the conservative Independence party. "Also they said they would never raise taxes, which persuaded lots of people to vote for them, then put them up. They're just another old-fashioned socialist party."

Jón was also attacked as a porn user after he got bored with the banality of the 400 interviews he has given since taking office and, asked by a French journalist to name his favourite website, joked that it was a porn site. The father of five, an emotional and open character, finds the smears upsetting.

Much of the opposition is in a newspaper edited by David Oddsson, an admirer of Margaret Thatcher who unleashed the bank boom while prime minister. The Best Party tactics are to stay silent under fire, remaining positive and courteous. ''It is a challenge, to be honest," admits Jón. "But if opponents say 'you are an asshole', I ask them not to state it as fact, but as their opinion."

Now journalists have identified a creed called "Gnarrism", defined as politics by people opposed to politics. Meanwhile the party is attracting interest around Europe, travelling to Austria, France and Ireland to explain its tactics, with a trip to Frankfurt next. Its members' dark humour doesn't always translate: Einar Örn said Irish activists had looked baffled when advised to listen to Lady Gaga.

The Best Party's popularity has waned as it tackles tough issues, but still one in five voters backs it and Jón is viewed as the nation's "most honourable" politician. The big question is whether it now moves to the national stage. Gudfridur Lilja Grétarsdóttir, head of the Left Green parliamentary group, is among those hoping it does. "We needed something to shake up the system, since it's so corrupt," she said. "Who knows if this experiment will work, but at least there is hope of change in Iceland. Without [this party], there would be none."