Note that I'm talking the board/strategy/etc type games, not including sports (so soccer, basketball, etc don't count here).
Criteria I'm using are:
Popularity
Longevity
Fun to play ranging from casual to expert players
Scrabble:
Great combination of words, math, and strategy.
Poker:
Great combination of psychology, math, and strategy.
Chess:
Perhaps the greatest combination of strategy and tactics.
Others I considered but didn't quite make the cut IMO were Go, Monopoly, and Risk.