@FarmerLenny answers:
My favorite game that is strictly two-player is probably Jaipur. I love the trading element and the constantly changing strategies based on what your opponent does. It moves quickly, offers good choices, and it's a game that really engages both players. If I'm thinking in general of what game I enjoy the most when there are only two players available, I would probably say Dominion. Dominion is a great two-player game. Some of the attacks are less nasty (notably Torturer--he can't stack as easily), but this might be a good thing. The experience is very different from a game that involves more players, but it's still excellent fun with two. In fact, I think the good tension is higher in a two-player game than a game with three or four players.
@Futurewolfie answers:
I actually don't own any two-player only games. I mostly get games to play with my board gaming group, which means that more players is often better than less. However, it is even better when games support two players, and not in an awkward, work-around variant that isn't very fun. I play two-player games most often with my wife (and second-most-often with my nemesis above), and the game that hits the table most often is Dominion. As @Farmerlenny mentioned, it's a different experience from four to two players, but the two-player game is still tense, competitive, and fun. Carcassonne is another great two-player game that hits the table a lot.
I've found that Tobago, though supporting two players, is a lot better with three or four (although two-player was significantly more fun after we knew what we were doing). Another game that supports two players is Race for the Galaxy, but while I enjoy it, I think it is significantly more fun with three or four.
I'm always looking for great two-player games (especially if they support more), so leave your favorite two-player games in the comments!

