Almost all games in real life are based on a finite set of integers (for example, we can index each possible movement in chess by an integer). But it is still pretty interesting to think about games that goes beyond the integers, to the uncountable, for example the Banach–Mazur game, which depends on the topological structure of the underlying set. We can play this game on $\mathbb R.$
However, to judge the winner, we have to move countably many steps.
My question is: does there exists some "uncountable" or "topological" games which can actually be played in real life? After all, many infinite things have finite analogs - so maybe it can somehow made to work in real life?
EDIT: Here is what I mean by the phrase "finite analogs" above. Of course with our countable language we cannot play games on an uncountable space. But if a nice finite or countable analog makes sense, then it would be great.
How about this:
Take an irregular shaped (bounded) region in the plane. Players take turns placing disks (considerably smaller than the region) into the region. Once a disk is placed, it stays in its position. Disks must lie completely within the region, and disks may not overlap. The winner is the last player to successfully place a disk according to the rules.
Because of the free placement, there are uncountably many game trees/game positions. However the game will end after finitely many moves, assuming that all the disks are the same size--or at least that the disk size doesn't go to $0$ as the game progresses.
You could play this game with other shapes than disks.
[See also Sam Loyd's 'egg game': http://www.jwstelly.org/CyclopediaOfPuzzles/PuzzlePage.php?puzzleid=Pz169.1 ]