RPG, Third Person Shooter and you're missing Survival Horror my personal favorite.
Mix TPS with either Survival Horror or RPG and you get epic combinations like Dead Space or Mass Effect.
RPGs with simulation aspects (think Persona 3's daytime events) that require you to create strategies to allow the player to actually defeat bosses and FOEs instead of defeating them by just spamming the attack option.
Just so you know: RPGs stand for RolePlaying Games, making RPG games redundant. Just mentioning this because it annoys me. XD
Fighters, but by a tiny bit. Action games like Ninja Gaiden (the Itagaki ones), Bayonetta and God of War are the top of my list. Along with a dose of Gears of War 3 and Diablo III for some variety.
My vote went to simulation. Why? Because there have been a handful of simulation games which I love to an almost obsessive degree. The SimCity and Civilization series have been my favorites for many years. Besides them, I still enjoy some of the classic Koei games, such as Uncharted Waters: New Horizons and Gemfire. My interest in Harvest Moon is a bit modest, as it pretty much started with the Gameboy games, peaked with Back to Nature (PS1), and ended with the DS games. In the last few years, I've become a serious Kairosoft fan, especially in regards to Grand Prix Story and Dungeon Village.
Apart from that, I continue to like puzzle games. My interest in RPGs began in 1989 and kept going through 2009, but unfortunately I haven't found one that's held my interest in the last couple years. With other people around, I can appreciate a decent action, sports, or party game. I occasionally like an adult game, but my interest has slowly veered towards worksafe romance games and any kind of visual novel.
I am ashamed that there is no shmup option on there. Yes, my favorite genre of game is the bullet hell shoot'em'up, or danmaku game. Bullet hell games such as Ikaruga, Touhou, Seihou, Mushihimesama, ETHER VAPOR, Hellsinker... the list goes on. Nothing is as fast-paced and difficult (besides maybe Beatmania IIDX and dancing games, which are essentially the polar opposite of bullet hell; instead of dodging everything, you're trying to hit everything). You have to remain steadfast and have rock-hard concentration, quick reflexes, and a touch of luck. What sucks is not being good at them, but hell they are some of the hardest games on the planet.
From there my likes branch off to all types of video games, but at the top are probably creative sand-box games, FPS, and music games.
I have a list of some, mostly ranging from the ones I'm good at to those I just like for casual play.
1) Action/Adventure or Hack and Slash: The Devil May Cry series first comes to mind, thank God for the recently released HD Collection!
2) Fighters: Can be based on old to new fighters, as long as they're good and simple to get used to before diving deeper into its mechanics.
3) RPGS: Good way to invest some serious time if I'm into quests and what not.
4) First Person Shooters: Not something I play for competitive reasons, but sometimes I enjoy the rush I don't normally get in other games and it's a nice change of what I'm used to playing.