From the comments, the desiderata are:
Preferably, a game that is/was in play by some human population (as opposed to one whose rules were written to have it fall in the complexity class that I am looking for), and one that involves only one player. Otherwise, if none can be found, preferably the game should have its rules written to resemble a "puzzle game" in that it should have a significant visual component (like a board/grid, visible game pieces, etc.) plus it should involve only one player.
Any "puzzle" from one of the EXPTIME-hard games should fulfill the requirements. For example, chess puzzles:
Chess puzzles are played by a lot of humans, only involve one player, and resemble (are) a puzzle game. The difficulty of finding the best move in a chess puzzle position should be exactly the same as the difficulty of finding the best move in a position in a 2-player chess game. So if we accept that finding the best move in a generalized 2-player chess position is EXPTIME-hard, then so is finding the best move in a chess puzzle.
Is it EXPTIME-hard to find the best move in generalized chess?
Probably? But I'm not 100% sure.
The EXPTIME-completeness result for generalized chess is Computing a perfect strategy for n × n chess requires time exponential in n, Fraenkel & Lichtenstein, 1981. The paper says that the following problem is EXPTIME-hard:
Given an arbitrary position of a generalized chess game on an n x n chessboard from our class of chess
games, can White (Black) win from that position?
In (generalized) chess puzzles, the question is a bit different:
Given an arbitrary position of a generalized chess game on an n x n chessboard from our class of chess games, where white (black) has a winning strategy, find a winning move for White (Black).
The first problem is at least as hard as the second problem, because you can use an oracle for the first to find a solution to the second in polynomial time: simply try all possible moves and ask whether the resulting position is winning or not. Then, once you find a move which results in a winning position, that move is the answer to the second problem. The first problem is EXPTIME-hard. I guess it's not instantly clear whether the second problem is as well, but it seems like it probably is.
If it is, then the argument from the Fraenkel & Lichtenstein 1981 paper also applies to generalized chess puzzles.
(Note that this doesn't apply to puzzles of the form: "Find mate in X for White", only to puzzles like "Find a winning move" (like puzzles on Lichess).