Another interesting class of games are multiplayer games that combine strategy and persuasiveness (the one I remember best from college days was Risk). A good Risk player has to accurately analyze which of the other players are the biggest threats AND make compelling arguments that lead to alliances or weaken the alliances of your primary opponent(s) at any given point in the game.
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Personally I like strategy games that combine strategy, economy, military etc. I like manager games (the more complicated the better) too.
In a multiplayer setting I used to play ikariam - that combines strategy/economy/military/diplomacy etc - back in 2008-9.
I got ranked #1 in score among ~3mn players back in October-December 2009, and then deleted my account in protest for some weird stuff that was going on with GameForge (the company that made it)... Some players were using bugs to climb the score ladder and despite that being against the rules (penalty for bugusing = ban), they allowed it. So I said let them have the bug-users there, legitimate champions weren't obviously worth their attention. And the game wasn't worth any more of my time and dedication if it rewarded cheating.
As for sports, which is another dimension, I prefer ...motor-sports and go-karts in particular. I find the challenge of shaving 10-20-50 msec in laptime very mental as one has to find and then go through the optimal lines, braking points, acceleration points etc.