To be honest, I love almost all types of games. Free roaming games have to be my favorite, as they give you the chance to explore a world that with any other game would be forgotten. It's almost if they're tying to say, "Take a look at the world you're trying to save from a townsfolk's perspective." A good example would be the Legend of Zelda series, most known for it's massive open world-ness and minigames .
Games with big storylines and a well-developed character also have to be considered my favorite. An alternative to this would be a game that pulls strongly on emotions, and causes you to somewhat cae about the world in which you save. An example of this would be Mother 3, one of my favorite games of all.
Games based solely on minigames are okay, however with something like WarioWare where each minigame is approximately four seconds long, that kills it. I want to be able to have a chance to enjoy a minigame. Not think on the spot for about a second before I have to do something.
Two-dimmensional platformers in my book are kindof iffy, but some exceptions come to mind instantly. Some would be famous ones like the original Sonic's and Mario games, while others would be around now like Cave Story or
Cobalt (play that), or freeware games like It Belongs In An Ancient Ruin! and King Arthur's Gold.
Multiplayer games can be okay, such as the aforementioned King Arthur's Gold and Cobalt, however games like FPS I can nevr get into.
And, the one genrte that I love the most is definately sandbox games, like GMod or Minecraft.