I've actually found there is a game from each genre I enjoy, at least for a while so there isn't any I outright avoid. However, there are some genres that I don't play many games from.
1. WRPGs (With the exception of KOTOR and Lord of the Rings The Third Age (Third Age had turn based combat very similar to Final Fantasy X which is one of my favourite games ever and is my favourite combat system ever), I've never really liked them, they're usually really dull and boring (especially Bethesda games) and the character creation system usually only makes people that look VERY unappealing to me and I find the overuse of choosing dialogue to get annoying (I just wish it would have a proper conversation in a PROPER cutscene, they're so much better, the people talk really strange like robots usually) however I think from what short time I played Mass Effect 2, I might enjoy that series a bit. Overall, WRPGs tend to have very little appeal whatsoever to me with very few exception mentioned above but even still, they didn't really amaze me.
2. Sports. Only ever played one with my family but, I actually enjoyed it but, I don't go out of my way to play them. Also sports really don't appeal to me in real life so being in the form of a video game isn't much better.
3. MMORPGs. I've tried several and while they initially interest me I soon get bored within a couple of days). I even tried playing World of Warcraft with friends but, even that got boring and I lost interest within a few days.
That's about all I can think of. So there is usually a game I like from every genre but some genres don't have many.