The final episode of “Late World” was a party to celebrate the show’s cancellation. “To me, it was very funny to play the part of the ungracious, ungrateful host,” Galifianakis told me. “I didn’t want to be there, so I decided to use that as my material.” When the reality of what happened began to sink in, however, and new roles proved scarce, his equanimity began to suffer. “Once you have a show, and it fails, people see you as a has-been, and that’s it,” he confessed. “The general attitude, from everyone around me, was ‘You blew it.’ I just had that feeling, like I was a wash-up pretty early.” Galifianakis spent the next few years performing where he could, taking the occasional acting role, even auditioning for game shows. When I asked whether the transition was hard, Galifianakis fell uncharacteristically silent. “When you go from having your own talk show to doing stand-up in a bowling alley,” he said finally, “you can react by getting mad or depressed, or by just going away, like people expect you to.” He took in a slow, thoughtful breath. “I reacted by growing my beard.”

Galifianakis’s beard has since become indispensable, both as source material for his jokes — “When you look like I do, it’s hard to get a table for one at Chuck E. Cheese” — and as a symbol of sorts to his burgeoning fan base of young hipsters, many of whom are defiantly furry themselves. It also seems to have made him funnier. Distinctive hair, facial or otherwise, is nothing new in comedy — Chaplin’s mustache, Andrew Dice Clay’s sideburns, Carrot Top’s carrot top — but in Galifianakis’s case, it actually serves as an organizing principle for his career, a useful dividing line between his formative and mature periods. The less hirsute Galifianakis of “Late World With Zach,” “Boston Common” and “Tru Calling,” was a talented wisecrack, a perpetual adolescent, smirking his way through the conventional stations of showbiz; the post-“Late World,” fully bearded Galifianakis plies much stranger waters, and he has adjusted his exterior to match. The beard lends him a subtly beatnik air, a link to the first generation of stand-up iconoclasts, like Lord Buckley and Lenny Bruce; at the same time,it gives him a professorial quality — dare I call it a gravitas? — that makes his more meatheaded material jarringly effective. Galifianakis himself put it more succinctly: “I look like a homeless guy now. People seem to appreciate that.”

One of the people who appreciated it was Kanye West, who saw Galifianakis perform at a Hollywood club called Largo in 2007. “I was doing a bit about how much I hate celebrity egos, and that seemed to resonate with him, for some reason,” said Galifianakis, with a barely perceptible grin. “He asked me to do a video for him, and I said yes, with one condition: I just go off by myself and shoot it, and he doesn’t get to look at it until it’s done.” To everyone’s surprise, West agreed. The resulting video, in which Galifianakis and the indie-folk icon Will Oldham drive a tractor around a cornfield while lip-synching West’s hit single “Can’t Tell Me Nothing,” has been viewed nearly half a million times. “I wasn’t sure what Kanye would think of it, to be honest,” Galifianakis said. “But his response was perfect, considering how we’d first met. He said it was the best video he’d ever made.”

The pint-size orange tractor that Galifianakis and Oldham frolic on in Kanye’s video sits at his farm in the mountains of North Carolina, not far from the Virginia border. When I met him there one damp, sunny morning in March, we took it for a spin — moving at a tortoiselike pace, perhaps to make the property seem bigger, perhaps because it’s as fast as the tractor can go. I balanced somewhat precariously on the tractor’s rear fender, and every so often a sudden lurch threatened to plant me facedown in the mud. “This tractor is painfully slow, but whenever you let your guard down, it screws you,” Galifianakis yelled over the whine of the engine. “My comedy is like that. You can put that in your article, if you want.” He was dressed like a gentleman farmer, in boots and a herringbone blazer, but his enthusiasm made him seem like a 12-year-old boy. “We’ve just passed through Lower Lickbottom,” he said grandly, gesturing at a clump of pines behind us. “And straight ahead is Pondoleezza Rice.”

An abundance of puns is not the only surprise in store for visitors to Galifianakis’s farm. Here in the country, his beard carries an altogether different set of associations — less “Pull My Daisy,” more “Deliverance” — and his demeanor seems to change subtly to match. His relationship to his home state is a complicated one (he refers to a nearby town, for example, as “the mouth-breathing capital of the world”), but Galifianakis, whose father moved to the U.S. from Greece at the age of 3, has a traditional side that the hill country brings to the fore. “I’d spend all my time here, if I could,” he told me, heaving a yeomanly sigh. “It would make my parents happy, for one thing. I pitched a sitcom once that would have been shot in the next town over, where they’ve got a racetrack for lawn mowers. It would have been a great show, based on stuff that actually happens around here.” I asked what happened to the sitcom idea. “I couldn’t get anybody interested back then,” he said, shrugging his shoulders. “Maybe these days I’d have better luck.”

Fortune does seem to be smiling on Galifianakis these days, but he has always been interested in seeing exactly how much he can get away with. The subject of race, a favorite since his stint as the effeminate redneck in high school, has remained a staple of Galifianakis’s stand-up, both because of his small-town Southern background (always fertile ground for comedy) and because of its status as the great American taboo. “It’s not a selfless thing,” he told me. “Wherever there’s something that people don’t feel comfortable talking about, that’s where the good jokes are. People might misunderstand you, but I decided, right after my show was canceled, never to dumb my material down for anybody. A bad comic follows his audience, catering to whatever they want; a good comic will always lead.”

Galifianakis is especially proud of “Apology,” a two-part bait-and-switch routine he developed a few years earlier. Several years back, performing at a dinner theater in L.A., Galifianakis delivered the following one-liner: “When I get drunk, sometimes my Southern accent comes out, and I say words like ‘y’all’ . . . and ‘nigger.’ ” It went over the way his jokes tend to: an instant of silence, then loud, slightly horrified laughter, sprinkled with the occasional boo. Then it became something more. “The audience was getting very upset — people in the crowd were yelling, ‘You’re a racist,’ ” Galifianakis told me. At this point, he interrupted his act to address a young black woman sitting in the front row. “I said, ‘I want to publicly apologize for this joke,’ ” Galifianakis told me. “I brought the woman up onstage, and I began to read an apology I’d written out, to the tune of Michael Jackson’s ‘Black or White.’ Halfway through it, another black woman came out from backstage, and we did an elaborate dance routine together. That was the close of the show.”