I was just looking over the rules for the Wizard, about how they get a new spell every level, and I thought: what does that mean for the Wizard as a character, as a part of the story. Does he suddenly realize how to do a new spell that he's been studying in his book? Does the power of magic now flow more strongly through him, enabling him to get greater power over it? There are undoubtedly lots of fishy answers we can use to try to make it make sense, but more often than not, we just don't question it -- he just gets the new spell and that's it. Dungeon World is a game at that moment, not a story.
I think I'd need a solid fictional explanation for what's going on with levels to do this game justice as a player or GM, or else... why not just ditch them entirely? Apocalypse World gets along just fine with advancements every 5 xp, but no measure of "levels" as such, why not Dungeon World too? Already, monster levels have been removed because they don't mean anything in terms of fictional positioning.
Are player levels so *core* to the old-school-dungeon-delving fantasy experience that they simply cannot be done without? If so, why not make them integral to the new-school-dungeon-delving fantasy story we still want out of the game?
Earthdawn, that old gem, called them "Circles," and it was something characters actually talked with one another about in the fiction. It was a measure of how powerful and renowned you were to say you were a "10th Circle Beastmaster," in their world, a bit like terms such as, "4-star general," "green beret," or "black belt" in ours.
I suggest we go one way or the other with Dungeon World. Put levels in the fiction or take them out of the game.