Probably some seamless world FPS or 3RD-person (one or the other, not sure, probably more 3RD-person to see our character) type of game. When I say seamless I mean it in a way that we've not seen in gaming so far, the technology probably doesn't exist for it yet, or if it does the GPUs would render it at 10 frames per second. And additionally it'd have a very detailed customization phase for your character.
There would be no loading after the initial "world loading" which I guess itself could take minutes, but after that it's done. You'd walk, run, swim, climb, fly... whatever, nothing would load anymore, it'd just be there (stored in memory or whatever the technicalities behind it would be). I'm thinking of a mixture of STALKER/Fallout mix for the setting, maybe not exactly post-apocalyptic... but probably occurring during the apocalyptic event(s), be it an asteroid impact, a nuclear war or accident, an alien invasion or some worldwide pandemic followed by social and cultural chaos and disorder (also have Children of Men in mind which also reminded me a lot of Half-Life 2's setting).
Well the setting itself and the game's context is unclear I'd still have to think about something, but the type of environment is, seamless, open-ended. And the game's "ending" (or more than one) might be your own, meaning that there's no scripted scene ending, you make your own and after all important tasks are completed the game's "main goal" is completed, the credits roll and you're back in the game and can continue playing, observing, experimenting or just firing rockets on abandoned buildings and see them collapse and explode in all sorts of physics fireworks imaginable.
That, or something very much story-driven, scripted to the bones, but with great story-telling, good character development, etc (thinking Half-Life series, Mass Effect 1, DAO, Baldur's Gate, Knights of the Old Republic, etc). Doing the story-driven type of game would be easier but I'd really want something truly technically and technologically seamless.