From various discussions I've had online and in person, I'd say RTS games. The degree that some of these games make you think is beyond almost any other genre, especially if you're aiming to be competitive. I love RTS, but I am by no means a genius, whereas I wouldn't be surprised if some of the avid RTS enthusiasts I've spoken are. Specific player bases I've spoken to are the Starcraft and Total War series, but there have been others too.
