His hour has turned into a year, and beyond that, who knows? To judge by the enthusiasm and undimmed curiosity of the 31-year-old, who was also awarded the cherished title of Comics' Comic by the influential comedy website Chortle, the sky is the limit.
"Curiosity" is a key word for Maxwell. It explains his drive. It started when he was a child and has never left him. In his show, he explained how members of his family joked about him being a weirdo because he would proffer endless facts about Mussolini at family functions: "They were baffled, amused and rather pleased to find this incredibly curious little boy in their midst."
Growing up as a Protestant in Dublin, Maxwell had a sense of being different from, but not at odds with, his neighbours: "It fired my imagination. But difference is the classic profile of comedians - gay, Jewish, adopted, black, small, tall, rich in a poor neighbourhood or vice versa."
Both of Maxwell's grandfathers were lay preachers, but he is circumspect about how much that influenced his chosen path. His parents certainly didn't stand in his way when he announced at 17 that he wanted to be a comedian: "I was the class clown - it was a natural progression."
There's little doubt that Maxwell experienced all the kinds of fun he talks about in his act, from "the craic" to "rooly booly" and "ri-ra", and back again. Armed with this, he started out at 17 on the comedy scene, which in Dublin meant Ardal O'Hanlon's Comedy Cellar. "I met Ardal, Dylan Moran, Barry Murphy, Alex Lyons, Dermot Carmody, Kevin Gildea, Joe Rooney. All those guys were 28, 29, and I was 18, they all had university educations, and I was a bit rough, they couldn't believe just how rough. I wore hoodies and baseball tops, much the same as I do now. They thought, who is this little car thief among us? But they put me on and were supportive when I came to London - I'd stay at Dylan's and Ardal got me gigs."
Among other comics that Maxwell namechecks for supporting him are Jimmy Carr, Brendan Burns and Ed Byrne. Byrne in particular gets high praise: "We're almost the same age, but Ed turned his talent into material success quicker than anyone. He became the benefactor of a whole generation of comics. We've all stayed at his house at low points in our lives."
Maxwell acknowledges that he was at just such a low point a few years ago, when he felt his career was going nowhere: "I was so disillusioned and felt I was being ignored. Comics found me funny; I went down well with audiences; but I kept wondering why no one was offering me more of a challenge than, 'Hey, do you want to be a TV presenter?'"
But his Edinburgh show last year was a high point, the culmination of the support he'd had from fellow-comics who had advised him to let his intellect run wild, and of building a new team around him, including Paul Byrne, director of This Is My Hour and his new show, Grand Royale.
This Is My Hour was Maxwell's third Edinburgh show, and it went off on some joyous political tangents as he did indeed let his mind roam free, assisted by a tricycle that he wheeled around. He was adept at targeting what he sees as reactionary Americans, and commentators such as Rush Limbaugh ("They aren't satirists; they're jesters"). And he goaded the Daily Mail lobby, too, suggesting, for example, that teenage mothers would save us from a demographic crisis: "They're paying for the older generations to come - we're all right because we have slags!"
Maxwell's passion for politics inevitably began at home. Growing up in 1980s Dublin meant that, while not being on the front line, he had relatives in Northern Ireland and was very aware of the sectarian troubles. The current global political climate naturally gives him no more cause for comfort: "I was in South Africa when the London bombs happened, and it was Muslim this, Muslim that. It's nothing to do with Muslims, it's a psychopathic billionaire who's waging a despotic war against working-class people all over the world." But as a comedian, he's aware that "if you try to be serious, your act risks turning into oratory. No one wants to look up to a clown."
"Clown" is Maxwell's preferred term for "comedian", a deliberate attempt to put his profession into perspective: "You can be thoughtful and incisive, but being the clown is what counts. If you're not getting laughs every 30 seconds, no one's going to listen."
And what about the title of the show? "I wanted it to reflect my experience in Kings of Comedy, but not overtly. I was in France when I came up with it, and in a French kind of mood. Grand Royale could mean lots of things - a casino, a David Niven movie, a cheeseburger. I wanted a title that would make me feel proud and be a gift to journalists; they're the ones using the title, not the audience."
Andrew Maxwell: 'Grand Royale', Pleasance Courtyard, Edinburgh (0131-556 6550), today to 29 August (except 9 and 16)