Big game. Deer specifically. I enjoy the challenge of hunting older bucks (for my region anyway) and the problem-solving that goes with it. There are so many different tactics/techniques that could be used, and even when you get halfway decent at it, the buck wins almost every time anyway. It's addictive.
And actually, I really am motivated by the problem-solving which is actually a big part of most big game hunting. I rarely shoot the little critters anymore, but as my kids get to hunting age I suspect I will do more of that again.