I'll be honest. Sometimes I want to play a game that makes me think on different levels, but most of the time I want to get in there and kill the bad guys/throw the ball/blow something up etc. etc. I like mindless violence. I play Call of Duty. I even like the story modes so I can sit back and relax while I kill things. The better it looks while I kill things, the better. So I love games like God of War and Dante's Inferno.
Game that show off a rich history also catch my attention. Not because I have to think while playing the game, but because the game makes me think about things that are actually important. Assassin's Creed was good for this. The game required more thought then my mindless shooters and my hack n' slashers and was still fun.
Game that require a lot of thought such as RTS or puzzle games often leave me feeling disappointed. They're fun while building up/figuring out the puzzle, but when I'm actually attacking I get bored. Or moving to the next puzzle is (Usually) boring. There are exceptions (See: Amnesia) but they're far and few between.
TL:DR - I like killing things and going herpa derpus behind my monitor. I don't want to think, most of the time, while playing video games. I just wanna blow stuff up. :flamethrower: