I think the first chess variant I was seriously interested in was Gary Gygax's dragonchess. It seemed like it'd be hard to get a set together, and gameplay'd be outlandish and complicated. And it is!
There are plenty of others, but I've played almost none of them. By far the most interesting, for a number of reasons, is Alice Chess. The pieces go through the looking glass every time they move. The most worthwhile is Fischer random chess or Chess960, invented in 1996 by Bobby Fischer.
Chess with different forces also looks interesting, especially Dunsany's chess. These are probably in the "a game or two would be fun" category.
There are a couple of four-handed chess games, aside from stuff like chaturanga and Enochian chess. I think the Dessau chessboard looks the most practical, as it eliminates pawn capture problems. Four-handed competitive play sounds more interesting than cooperative play. The Battle of Four Armies: capture an enemy king to force that player out of the game (their pieces cannot move or threaten, but are not removed from play until individually captured) and the last king standing wins.
The miniature shogi variants look like they give quick and elegant games. There are a couple of miniature versions of orthodox chess: HP Minichess, apparently presented by Marvin Gardner in Scientific American, is played on a 5x5 board. Toystore Chess is essentially miniature chess with placing. Quick Chess and Speed Game Chess are played on a 5x6 board.
The shogi variants are wild and impressive. Five-minute poppy shogi and mini-shogi are beautifully minimalistic, while taiyoku shogi is epically huge and unwieldy: there are 1296 squares on the board and each side has 402 pieces of 209 different types, totalling 253 different moves. Totally crazy.