I'll play anything and everything, though my absolute favorite is same-room/screen co-op.
Most of my friends have a different thought process than me, so co-op games are always interesting because of the varying play styles. Normally we're working towards the same goal, and we don't take the game too seriously. Working on my second play through of Lord of the Rings: War in the North same-screen co-op and loving it.
However, one of my friends I can't play anything with. He's just a troll and I can't stand that type of person. His idea of having fun is ruining the fun of everyone else, and It's way too immature for my liking. We can't even play a game like Little Big Planet together because he's soo intent on working against his partner(s). Hell, if we're on a team in a 2HG game of Magic The Gathering, he will play cancels on my spells just for kicks. I make it a point to play games with him as little as possible.