Many combinatorial games that are studied in an academic setting have very simple rules. Just because a game has a multitude of rules does not remove it from the combinatorial setting, however. For example, Rick Nordal sent me a link to his game, Connect Score.
This game uses printable boards of chess pieces, each given a number. The game proceeds as a mix of Dots and Boxes and Chess. Each turn, a player first chooses to add one line which is the boundary of one or two boxes, just as in Dots and Boxes. Unlike Dots and Boxes, you do not earn additional turns by closing off boxes. Instead, the chess piece in that box becomes activated. (All pieces begin inactive.)
In the second phase of each turn, the player gets to use all activated pieces and have them shoot at each other. The colors do not matter; each player can shoot with any of the active pieces. Any piece which gets shot is then removed from the game, and the current player earns the number of points associated with the "killed" piece. Those pieces can then no longer shoot. Pieces shoot in any order the current player wishes, and can shoot multiple times per turn. They do have to shoot in order, so two pieces cannot shoot each other in the same turn. At the end of any turn after the first piece has been activated, there will still be at least one active piece.
Each Chess piece shoots in a different manner, so they can't (all) target each space on the board. The rules for each are described on the website; I will not list them here as they are too numerous. Most of the pieces shoot similar to the way they can move in Chess, so it's not terribly difficult to remember if you've played some Chess.
After the shooting phase, the current player's turn is over and the game proceeds to the next player.
When all pieces are either active (and can't shoot each other) or removed, the game is over and the player with the highest point total wins.
Rick has produced a bunch of starting boards available on his site. Ernie and I found it fun to both play with a bunch of different pieces and one where each piece was a rook.
If you wanted to play this game not on paper, you could presumably use actual chess pieces, each on a stack of coins to keep track of how many points they're worth, and dominoes to denote the edges drawn. Sounds like it might be more epic, but also harder to wrap your head around! :)
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