Bill Bailey hates jokes – always has done. When somebody tells him one, he says, he feels as if he's being mugged – the buildup, the promise that he'll find this one funny, the forced reaction. Ugh. Nothing worse.

Bailey is possibly the world's least likely rock'n'roll comedian. It's not just the aversion to old-fashioned one-liners; it's the way he tells his non-jokes (cerebral shaggy dog stories that don't always conclude with much of a laugh) and his appearance (shoulder-length half-head of hair and mad, staring eyes, like something out of the Hobbit). But rock'n'roll comedian he is. He sells out stadiums, does world tours, has his share of obsessive fans and struts the stage with guitar in hand. He has a thing for progressive rock, and in a way his epic tours are like classic 1970s concept albums. Dandelion Mind, Tinselworm, Part Troll, Cosmic Jam: even the names of the shows evoke prog rock. In his latest, Qualmpeddler, he confronts his myriad doubts about the modern world via religious dubstep, folk bouzouki, Higgs boson, rare planetary alignment, Spice Girl reunions and more.

We meet at his Hammersmith office, which is every bit as bonkers as you might hope. Every object tells an unlikely story. There's the vintage clarinet-reed vending machine picked up in Philadelphia; a load of french horns that he bought as a job lot and turned into wall lights; a massive Bill Bailey Toblerone bar propped up on a grand piano (the manufacturer gave him mountains of the stuff for a piece he was doing about hoarding for the apocalypse); and a huge, terrifying bird.

What is that owl thing, I ask.

"That 'owl thing' is a Eurasian eagle owl." It's stuffed, but looks pretty lively at first glance. "It's a gift from my wife, because we rescued one from a Chinese restaurant last year." Where? "In China. So it's a reminder of that."

How did he rescue an owl from a Chinese restaurant? "We bought it. It was on the menu. It was a live owl, in a cage, on the menu waiting to be eaten, and we haggled with them for a bit and bought it for $400. We put it in a taxi, drove up to the woods, and let him go. I talk about it in the new show. Our son was with us at the time. He was eight then, and he filmed it on his little camera. There's this very sweet film I show at the end of the standup show, of us releasing this owl. The camera's a bit jiggly, then you see the owl emerge out of this carboard box, and then it flies off."

The stuffed owl in the office is massive. "Yes," Bailey says. "It is. It's the largest. My wife Kristin is a collector of oddities like this." Are you sure this isn't the owl that was on the menu? He grins. "It may be! I am quite tempted to bring it out at the end of the show and say, 'We changed our minds! We were going to release it, but we thought no, fuck it, it looks much nicer in the office.' "

This being Bailey, he's not quite finished yet. There's a bit more knowledge to impart. "The Eurasian eagle owl is an apex predator, so you can release them anywhere in the world and they would be fine. An apex predator means that nothing preys on it." There follows a quick class on apex predators, humans, jaguars and the South American jungles.

He has an astonishing ability to soak up obscure knowledge and incorporate it into his act. As a boy, he says, his mind was a sponge. He grew up in Bath as Mark Bailey – he was nicknamed Bill by his music teacher, becase he played the song Won't You Come Home, Bill Bailey so beautifully on his guitar. He was an only child, to a doctor father and nurse mother, and spent much of his time chatting with adults, mining them for knowledge. He sees his own son, also an only child, doing the same thing now. He'd tell tall stories, and often wonder how he'd dreamed them. "You spend a lot more time on your own as an only child. And there's space to allow your imagination to take flight. If you're bouncing ideas off someone, you take it one way and they take it another, wherereas if it's just you, there are no constraints."

Was he ridiculously clever? "I won't lie to you," he says. "At school I found it easy. Every year I got the prize." Was it a posh school? "Not particularly – it was direct grant. All boys. And every year for the first three years they said, 'OK, the English prize is Bailey', then geography, history, maths, physics, everything. One year I won them all."

His fellow pupils must have despised him. "They did. 'Ah, you bloody swot.'" It gets worse, Bailey says. He wasn't simply brilliant academically – he was great at everything. "One year I was captain of the cricket team, and the lead in the school play. Fair play to the English teacher who decided on The Resistible Rise Of Arturo Ui – good on yer." Bailey's never been backwards at coming forwards: even today, his promotional material stresses that he's not simply messing about with comedy music – he's pitch-perfect.

Did he hate himself for being good at everything? "I sort of did. I rebelled against it, and by the fourth and fifth year I just went off the rails." He stopped trying, stopped winning prizes, and joined a band. Going off the rails is all relative, though – he still managed to get an A in his English A-level. He started studying for an English degree, but gave up after a year. "I was restless. I wanted to get on, be challenged a bit more. I suppose there is a natural sense of rebellion somewhere along the line. At school, I was bored with the teachers, and there were moments where I felt they were singling me out… I suppose because they thought, 'Oh, here's a student we can get great results with', and I resolutely refused to go along with it. The same at university. I just got bored with it."

For a decade or so, he drifted, travelling across the globe in search of the meaning of life (spending lots of time in Indonesia, where he's off to later this afternoon), following any hippyish trail he came across, but also working – touring with a theatre company, playing in bands, doing comedy.

Back then, he even told jokes. Lots of them: short and punchy, the ones he can't stand. He was part of a double act, working at the Joker Comedy Club in Southend. One night in 1994 the owner rang to tell him the closing act had dropped out and asked if Bailey could do an emergency shift.

"But my partner Martin was on holiday. He said, 'Can't you just do it on your own?' I went, 'I don't know if I can', and he said, 'Well, here's the money', and I went, 'Oh, I get the same money if it's just me! I'll do it."

He did the gig, and it was disastrous. "The Joker is quite raucous: they like their comedy and they let you know one way or the other. Then some bloke said, 'Oi, mate, tell us a joke!' I thought, 'Oh my God, I can't think of any jokes at all.' So I went, 'OK, three blokes went into a pub', and straight away there was attention because this was something the audience could respond to."

The problem was, there wasn't a joke at the end of it. He didn't have a clue what he would say next.

"I'm thinking, 'I'm so in the shit here.' So I said, 'I say three – it was probably more like four or five', and that got a couple of laughs. Then I said, 'Well, I say five – it was 10. OK, 10 blokes go in a pub', and I say, 'OK, there were quite a few more than that – 20-30 blokes', and then it was 100, 200, a village, a small town, an area of Holland, northern Europe, and it went on and on, extrapolating this thing from three to most sentient blokes. 'OK, the entire male population of the world goes into a pub and the first bloke goes up to the bar, and says, I'll get these.' And I said, 'What an idiot!' And it got a huge laugh, and I thought: that just came out of nowhere."

It was a revelation. Bailey realised that was what he actually enjoyed – deconstructing humour. And if he could get a laugh in the process, so much the better. "What I'd been doing up till then was not really what I'd been wanting to do. It was like a facsimile of a comedy act: it was what we'd seen other people do. It was jokes and one-liners and songs, but it wasn't really what I wanted to do. I'd settled for it."

He thought if he'd managed to dismantle a joke in front of such a tough, beery crowd, he could develop this. So he went on to tell jokes set in the 1950s, in Shakespearean and Chaucerian times, about all sorts of unlikely subjects – and the skill of these anti-jokes lay in the telling rather than in the punchline, which often didn't exist anyway.

What happened to poor Martin? "Well, he came back from holiday, and we did some more gigs, and then I said, 'I want to do this on my own now', and he said, 'Yeah, all right, fine.' We parted amicably. He went on to work for a photo archivist, which suited him down to the ground, because he was obsessed with cataloguing."

So Bailey had become an accidental surrealist. And after a long wait, his career took off quickly. In 1995, he won a Time Out comedy award; the next year he was shortlisted for the Perrier at Edinburgh. His act was more wordy than Proust, his humour was clever and gentle, most of his non-jokes were accompanied by music, and he appealed to people who liked to think of themselves as being on the smart side. He was also virtually impossible to categorise. I ask if he can explain his humour. And he's stumped.

Bailey in Edinburgh in 2003, with his son Dax in 2011, and in Black Books. Photographs: Rex Features; Getty Images; Channel 4

"Well …" He takes a deep breath. "It's a mixture of different elements: straightforward standup, observational, a bit of politics, a bit of philosophical discourse, a bit of interaction with the audience where I try to subvert the nature of audience participation. Then there are various musical sections of the show, where the music is observational, so it's not a parody per se: the music is the joke. So, for example, I bemoan the fact that the church has lost its influence in society – the church was always a big part of my childhood; now people don't take it seriously. The church has lost touch with the youth, they need to try to re-engage, so I imagine writing a piece to promote the church using dubstep, so we have an organ section, which turns into dubstep, which turns into an animated film in which the Archbishop of Canterbury turns into a ball of hair, which kills the devil with lightning... Hahahaha!" He's laughing not at the sketch, but at the impossibility of explaining it. "I can see your problem."

Has he ever tried to break it down like this before? "No! No! I'm trying to draw the elements together." Cognitive dissonance apparently comes into it somewhere – "something we all experience, but we maybe don't know we do".

Now I'm having to ask him for an explanation of his explanation. What's cognitive dissonance? "It's when there are two opposing thoughts in your head which seem to cancel each other out, yet you're able to hold them simultaneously. Smoking is a classsic one. 'This is bad for me; I like smoking.' You recognise the dissonance in those two thoughts. So I say to the audience, 'You all know what this is', and they think they don't, but they do. That's what I like to do," he says with a click of his fingers, as if he's just got it. "I get people to realise that they know all these things."

Because he's so mellow most of the time, it's easy to forget that Bailey is a political comedian – or at least a comedian with politics. Once, when he was asked what he thought about Tony Blair, he found himself on the front pages. "I just came out with some involuntary eruption. Among the general ranting, it came out as 'warmongering legacy junkie'. It could equally have been 'legacy-mongering war junkie'." But, he says, his contempt for Blair is nothing compared with that for today's government. When he turns to George Osborne and David Cameron, he's almost incoherent with rage. "This is different. This is more like a horror, a deep horror: to see how this confidence trick is being played out in front of everybody. It's evil genius. The hideous, two-headed Osborne-Cameron creature just frightening you with the utterly amoral … It's almost brilliant, the way they have turned our justified and rightful anger against bankers on to 'benefit scroungers'. How do they do that?" He comes to a stop, bewildered.

There's something Reithian about his humour, I suggest: inform, entertain and educate. "Yeah! Yeah! I do have that," he says enthusiastically. "I want it to be more than you just come and have a laugh. Obviously, first and foremost, that's what it's about – people pay their money and they want to have a laugh and take their mind off the fact that the high street is closing down and their Jessops vouchers are no longer valid. But there's also a bit of wanting people to come along and go away with something. 'You know what? I know what cognitive dissonance is now. I know who wrote this piece of music.' "

Bailey has thoroughly enjoyed his success, not least because it made his parents happy. For a long time, they had patiently watched him drift from continent to continent, dabbling in the performing arts. Even though they never said as much, they must have wondered what on earth he was doing with his life. After all, they were practical people with serious jobs and a Protestant work ethic. "I think they were concerned, more than anything, because comedy seemed to be a very uncertain future. Then they heard me on Radio 4, and they were like, 'Oh, that seems to have a bit more substance to it.' "

His mother died in 2005, but his 80-year-old father still enjoys his shows. ("We have got a similar humour. He's into surreal, odd stuff.") As does his nine-year-old son, Dax, who has already started heckling him at gigs. He says Dax is so used to seeing him introduced on stage that whenever he comes across a microphone, he feels compelled to do the honours. "We were on holiday recently and found this karaoke bar, and he just took over. He had the mic and he was like, 'OK, we're going to take a short break and we'll be back in 10 minutes,' then he introduced me." Bailey says he was so taken aback that he couldn't think of anything to do but perform, so he and Dax did a duet.

Bailey met his wife in 1987 at one of his gigs when he was a jobbing comedian. Kristin, who is now responsible for his business affairs, was running a bar in Edinburgh. He wooed the former costume designer with a weekly letter for 12 months till she finally weakened. In 1998, they married on a whim in Banda, Indonesia. Dax is named after an Indonesian friend. The three of them share a house down the road from his office with four parrots, two monkeys, a python, a chameleon, a few fish, two starlings, a number of cats and five rescue dogs (three of which he had shipped over from Indonesia).

Now 49, he had to wait until he was 31 for his first sniff of success, but before long he was omnipresent – and not just as a comedian. There was Bill Bailey the actor (notably alongside Dylan Moran in the TV series Black Books), the panel show fixture (pop quiz Never Mind The Buzzcocks), the ornithologist (Bill Bailey's Birdwatching Bonanza), the natural historian (Baboons With Bill Bailey), the music instructor (Bill Bailey's Remarkable Guide To The Orchestra). And now there's Bill Bailey the champion of forgotten heroes: he has just made a two-part documentary about Alfred Russel Wallace, Darwin's colleague and rival, and co-originator of the theory of evolution.

Bailey loves doing the non-funny stuff. "I was in Jakarta recently doing a talk at the Indonesian Heritage Society about Wallace … There were no laughs in it. It was just a talk about something I felt passionate about. There were a couple of nods of appreciation, that's all, and I felt great at the end of it. Quite satisfied." He loves to escape the confines of comedy, the tyranny of the laugh.

As he's talking, I find myself staring at a creepy Bill Bailey mask. "Ah yes," he says with a smile. "I was headlining the Saturn stage at Knebworth, and headlining the Apollo stage was Slipknot. I thought, because Slipknot's trademark is scary masks, I'd have my own Slipknot mask made of myself." He tries it on, and he does look genuinely scary. But not for long.

I offer him a Malteser. "But I was just about to offer you a Belgian chocolate biscuit. I'll see your Malteser and raise you a Belgian Titbit."

There's something heartening about Bailey's success. After all, comedy can be such a ruthless, backstabbing world. Do you have to be a show-off to get on? "Undoubtedly. I don't think any comic could say there isn't a bit of them that doesn't want to show off. You have to have that, otherwise you can't really do it." Do you have to be tough? "You have to have a thick skin, yes. If you're going to do something as foolhardy as standup, you've got to be able to take it on the chin if someone has a go at you." Is he really thick-skinned? "Probably not. But I am resilient."

It's easy to imagine Bailey as a much loved university lecturer – perhaps professor of music, popular entertainment, natural history and joke-free comedy. Has he ever fancied an alternative career? "No, because the stage allows comics a certain freedom of expression that you don't get anywhere else. You can say whatever you want – it's the last bastion of free speech in some ways. So I feel there's an obligation to make it that bit more interesting …" He clicks his fingers impatiently, searching for his words. "A bit more about something, a bit more challenging, for me and the audience."

Bailey puts on his coat on and says he's got to be off now – he's going straight to Indonesia. How's he getting there? "The 209. Goes all the way to Jakarta!" That sounds dangerously like a joke. He apologises, locks the door behind him and heads off down the road to south-east Asia.

• Bill Bailey's Qualmpeddler is touring from Friday until 3 November; go to billbailey.co.uk for details. Bill Bailey's Jungle Hero airs on BBC2 at 8pm tomorrow and next Sunday.