Monopoly, because it can turn into a marathon session. Endless outcomes, and whoever's "banker" has to keep her/his mind sharp.
Clue, because I'm never really satisfied with the outcome. It should always be Miss Scarlett in the Conservatory with the lead pipe. That girl is so damn guilty. But the game shuffles the culprits every time to keep you guessing. And I like using the secret doorways.
Go to the Head of the Class, because I'm a smarty-pants and like to show off. I never play with anyone as smart or smarter than I am, because I have to win this game. I mean it. I will always be Head of the Class. Or game pieces will fly.
And you, Professor Plum? What would feed your game-playing soul? (Board games only -nothing that needs batteries.)