So, with my brain still fried and my cat slowly drooling on my lap, I'll give a brief rundown of the types of games that I am most drawn to- this is nowhere near comprehensive, just a brief summary of my very favorite game types.
For me, I think it really started with Spyro the Dragon. Looking back now, Spyro appealed to both my love of exploration and fantasy. My first career choice as a kid was becoming a dragon, and while I couldn't achieve it in real life, Insomniac made it possible.
Favorites - Spyro the Dragon (1&2), Jak and Daxter, Kingdom Hearts (1&2) Shadow of the Colossus, Portal, Mirror's Edge
RPGs (turn based and realtime)
I almost put JRPGs up there, but that isn't entirely accurate. I'm very fond of strategic, (usually) turn based battle systems, and having multiple or changable party members at my disposal with skill trees/upgradable stats. An over-exposure to anime and manga going through junior high and high school did nothing to broaden my exposure to other types of RPGs.
Favorites - Final Fantasy 7 (I'll write an entry on this later), Other Final Fantasies (Tactics, 12), Digimon World (3, Dusk), Monster Rancher (3, 4), Chrono Cross, Dofus, The World Ends with You, Persona 4
I'd love to hear suggestions on games that I should check out, as I've always had a pretty limited budget, both for time and money, and I know I've missed out on a lot of good games. I've also missed a lot of amazing classics, as people frequently remind me. Partially because I'm relatively young, and partially because gaming was not my primary leisure activity until very recently.