Really, I find myself attracted most to games that antithesize the triple A games industry. That isn't to say I don't enjoy certain big-budget games, but I find myself more and more interested in games which don't necessarily involving shooting, killing, and fighting in any way. Probably my biggest attractor to a game is clever puzzles. Games which let me think, and give me a real sense of accomplishment from the completion of each task, due to the fact that I had to think. If the art style is unique, with decent art direction, if the soundtrack is nice an atmospheric and if the gameplay is a good reflection of the art direction, I'll be more inclined towards the game as well. I'm continuing to find that I pick up a shooter or a fighter or a third person action game, and get bored pretty quickly, and continuing to find that point-and-click adventures attract me more.
If we're talking about a weakness as in something that causes me to break down, within games, it's sandboxes. I can't deal with sandboxes. The reason is that I have a long list of games I have to get through, and it continually grows as new games come out. When I'm playing a game, if I'm not immediately invested in it, I find myself stressed about how much I still "have to play". The thing is that I get stressed out if I'm not doing everything available to me in a game, because I feel like I'm missing out on a lot, but I also get stressed out when I'm not progressing fast enough in a given game, so I end up just frustrated playing any kind of open world game. Most recently it happened with Assassin's Creed 2, before that it was Arcanum... I guess I just like linear games more?