I've tended to gravitate toward survival games, though I also like MOBAs (Dota 2 and Deadlock specifically, I bounced off LoL despite trying), an old MMO that seems like it hasn't had new players in a decade, roguelikes (traditional, berlin-ish type), and a smattering of games from a few varied genres I don't usually tend to play a lot of.

Games I tend not to enjoy are things like competitive FPS, rhythm games (they hurt my hands and make me feel bad for being quantifiably inferior to other people), sports games (just not big on sports), and fighting games (for similar reasons as with rhythm games, but with the added suffering of live PvP).