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.

One example is the ring game, which is defined here - in essence, the game (or a slight variant thereon) can be described as:

Start with a Noetherian ring $R$. At each turn, replace $R$ with a quotient thereof. If a player can make no legal moves (i.e. $R$ is a field, thus has no proper quotients).

One can notice that if we play this on $\mathbb Z$, then this is equivalent to the game $*\omega$, since the first move takes it to $\mathbb Z/n\mathbb Z$ which is clearly equivalent to $*k$ where $k$ is the number of (not necessarily distinct) prime factors of $n$. However, as evidenced by certain unanswered questions, the structure of the game can get fairly complicated when we consider more complicated rings. One could generalize to replace "ring" with any algebraic structure they desire - like one could play on groups with no infinite ascending chains of normal subgroups.
