A recent XKCD comic charted the difficulty of various games for computers, from Tic Tac Toe and Nim being solved for all positions, to computers mastering the physical game of Beirut and mental game of chess (the 2006 Deep Fritz vs Vladimir Kramnikin games, previously). There are other games that are basic on the face, but whose potentials for move combinations is so vast as to be beyond the scope of computers. Marion Tinsley was the last great human checkers player, matching off against Chinook in the last 6 games of his life, each ending in a draw (previously). Checkers was finally solved in 2007 (Google quickview; original PDF), and is largest game that has been solved to date, at 8x8. Solving Othello might be possible, if the decision tree were truncated, as the 10x10 board game tree complexity is very huge. The 19x19 Go board is is often noted as one of the primary reasons why a strong program is hard to create, though some programs are getting better at optimizing move evaluations. More: computerized gaming solutions previously, and the Wikipedia page for solved games.
"It was my luck (perhaps my bad luck) to be the world chess champion during the critical years in which computers challenged, then surpassed, human chess players. [...] What if instead of human versus machine we played as partners? My brainchild saw the light of day in a match in 1998 in León, Spain, and we called it "Advanced Chess." Each player had a PC at hand running the chess software of his choice during the game. The idea was to create the highest level of chess ever played, a synthesis of the best of man and machine." The Chess Master and the Computer: A article/book review on computer chess and the state of the top-level chess world by Garry Kasparov. [more inside]
Who was the most dominant athlete of all time? If athletes include draughts players then Marion Tinsley makes a good candidate, losing but 7 games plus 2 more to a computer over the course of a 45 year career. [more inside]