I mostly find myself playing: Sport games, survival, racing games, mecha games, fighting games, dating games and occasionally an MMORPG/RPG.
Sport games are nice because I can play it at my own pace if I am playing a sort of franchise mode, where you own a team or something or simulate being the player.
Survival games like DayZ are huge to me. I love L4D2 or games like Fallout, but an open game where you just try to live is the funnest to me. I don't need a plot as long as the game takes some skill to play.
Racing games are what I grew up on. I've played almost every major driving game since
Galaxy 5000 for the NES. GT, Driving Emotion Type S, Ridge Racer, Forza, Wipeout, Burnout, etc., etc.
Mecha games are my big crush. I adore the Armored Core series. Just wish they'd finally upgrade the single player campaign to an actual detailed story.
Fighting games are another big love from being a teenager. I've always been an MVC2 gamer. Tekken I use to get smashed on by my friends who are godly at it, so I kept with the fast paced, combo oriented class of games. BlazBlue was my more recent love.
Dating sims, use your imagination.
MMO's I enjoy. All RPGs I enjoy. Chrono Trigger is my favorite game of all time, and I recently finished up a few binge months of Tera. I like the MMO experience, but hate tab target. Kinda why Tera stood out.
Oh. And music games. Who doesn't like Ouendan?