It's more important to me than it should be. I have an obsessive personality, and seeing that I'm missing 1% or something like that will bug the SHIT out of me. It's like a puzzle that's missing a piece. It's like someone who keeps singing one particular note a quarter-tone flat, and no one else seems to notice.
It's kind of a pain in the ass, too, because going for 100% completion is almost never fun. I wish they'd just leave it out of games, in most cases.
100% completion usually comes down to collecting a bunch of random shit. Sometimes that's fun, like when exploring is a natural part of the game world and there's some reward to going back because you've gained new abilities that let you open areas you couldn't open before (like in Super Metroid). It's usually not fun when you're just collecting random shit scattered around the game world that you've already navigated a million times, like flags/feathers/etc in the Assassin's Creed series. If you're going to let us buy an in game map showing us the location of those things, it's at least tolerable. If you're not, that means I'm going to have to look up a guide of it on the net, and constantly alt-tab between that and the game while I collect said stupid shit. I HATE doing this.
Sometimes it means completing random challenges and minigames. That can be fun if you can complete them in a few tries and they give you a chance to do something you don't normally do in game, or use some existing ability in a new and innovative way. When those challenges and minigames are absurdly, inexplicably difficult, it's not fun anymore.