(No spoilers in this post. Just mentioning it because one part looks like it might veer into spoiler territory.)
Puzzles, platformers, and classic-style JRPGs. A lot of my favorite games combine the "puzzle" and "platformer" genres in some way - eg. Wario Land, Oddworld, etc. (There are limits though. If the puzzles get to the point I'm having to frequently consult a guide, it's a bit off-putting. La-Mulana 2 is an example of this; even the first was pushing it a bit in that direction, but the 2nd was well and truly too far for my liking.) If I had to name a single favorite though, it's from the JRPG genre, namely Final Fantasy X - I suspect many of you knew that either from me mentioning it in the past, or from how closely my RPG's gameplay is based on FFX's.
I can enjoy most of the common multiplayer genres - FPSes, RTSes, fighters or racing games - on a casual level for an occasional game, but I'm not particularly into them and definitely not all that good at them.