I just want to make a comment (which is too long for an actual comment):
We could also play this game with groups. One starts with a nontrivial group $G$. A move consists in replacing $G$ by $G/\langle\langle a \rangle\rangle$, i.e. we mod out the smallest normal subgroup containing $a$, which should be some element with $1 \neq \langle \langle a \rangle \rangle \neq G$. Groups which lead to a finite game are precisely those which satisfy the ascending chain condition with respect to normal subgroups (do they have a name?).
Actually we can play this game for every algebraic structure: Given a variety in the sense of universal algebra, we can also play the following game: Start with with a algebra $A$ in the variety which has at least two elements. A move consists in replacing $A$ by $A/a \sim b$, where $a,b \in A$ with $a \neq b$ and $A/a \sim b$ has at least two elements. The terminal position is the terminal algebra which has precisely one element. An algebra yields a finite game iff its partial order of congruence relations is noetherian.
Perhaps $\mathsf{Ring}$ doesn't provide the easiest variety, and one should try other examples first. Finite sets are boring, but what about about finite abelian groups?