Big List of examples of recreational finite unbounded games What are some examples of mathematical games that can take an unbounded amount of time (a.k.a. there are starting positions such that for any number $n$, there is a line of play taking $>n$ times) but is finite (every line of play eventually ends.) Also, it would be nice if it were recreational in nature (by this I mean I basically mean its nontrivial, I could conceivably be enjoyed by humans theortically.) One answer per game please, but edit variants into the same answer.
This is a big-list question, so many answers would be appreciated.

Higman's game$^*$
Using finite words on a fixed finite alphabet, two players take turns, in each turn specifying a word that contains no already-specified word as a subsequence. The game terminates when a player (declared the loser) is unable to specify a word, and the other player is declared the winner. 
By Higman's Lemma, every such game must eventually terminate; nevertheless, the playing time is arbitrarily large, depending on the initial words chosen.
$^*$ I'm making up the name and the game. Similar games with much longer finite potential playing times could involve specifying trees or graphs (instead of words) that avoid homeopmorphic embeddings, and appealing to Kruskal's Tree Theorem or to the Robertson-Seymour Theorem for proof of termination. These are all examples of the following generalization:
WQO games
More generally, suppose $(X,\le)$ is any well-quasi-order (WQO). Now if players take turns specifying elements of $X$, then eventually one of the players cannot avoid specifying an element $x_j$ such that $x_i\le x_j$ for some already-specified element $x_i$ (i.e., $i<j$), thus terminating the game. The playing time is always finite but arbitrarily large, depending on the initially-chosen elements. 
