It depends entirely on the game and what I'm in the mood for at the time. If the gameplay itself isn't any good, I generally won't put up with a game for very long, but as long as it's at least mediocre, I've been known to tolerate it for the sake of a very interesting story/setting (e.g. Korsakovia) or because I really like some aspect of it like the art style/design (e.g. American McGee's Alice).
Ideally, though, the best part of a game I'm playing will be the gameplay itself, whether because it's very polished and fun or because it's a novel idea that I haven't seen before. It can be almost any type of gameplay, as long as it's well done and/or creative enough, whether it's a twitchy action game or a turn-based game that involves more time thinking about what to do than actually doing it.
If I really, really like a particular game, I'll keep playing it to get some achievements if the process of getting them is enjoyable (as opposed to the tedious ones like "spend the next six months of your life killing 18 million of Enemy X") or try it on a higher difficulty level, but not just for the sake of doing it. Games like that are the exception, not the rule, and I have to like them for enough other reasons to want more of them instead of moving on to something new. That does not happen very often these days with my backlog of games I'm trying to catch up on.