Hmmm. Good question.
I guess my list would look like this:
Excitement: I want to be excited by the game I'm playing. I should really focus in on it, it should affect my heart rate, and my BP.
Discovery: I like it. Whether it's exploring the map, or discovering the secret of the temple, or what. I just like it.
Challenge: My number three. I love a game that challenges me, requiring that I learn a new skill. This will probably be moving down the list, in years to come, as my twitch speed drops. Old age is a Bichon Frise.
Destruction: I like the realism of a destructible environment, in both single-player, and multi-player games. If I'm hiding behind a car, and someone fires an RPG into it, then I'm either now hiding behind some other piece of cover, or I'm toast.
Completion: Some games I love have a story arc, like Dying Light, or the Fallout games, or the Far Cry games. Others, like the Battlefield games, don't. I like games with a story arc, but sometimes you don't want to have to bother with an arc.
Story: Goes along with Completion. A good story can really make a game. But it's sometimes not needed. Like with the Lemmings games.
Competition: Like you, Polly, I grew up on mostly single player games. But, I do love the Battlefield series. So, not at the bottom, but down a few from the top.
Strategy: I enjoy RTS games, especially the StarCraft games, but FPS is my first love.
Fantasy: I like a good fantasy game, even though I mostly prefer FPS games, which usually aren't fantasy.
Community: I'm too much of a misanthrope to like community, really.
Designing: Not a SimCity fan, or other similar games. Just not my style. Although I do like the Rail Road Tycoon game.
Power: Being OP is boring.