When learning a new programming language, I like the idea of having a familiar program or algorithm to reimplement in the new language. My personal favorite is a little game that I prosaically call 543. I have no idea if there is an actual name for this game as it was taught to me in a bar. 543 is a two-player deterministic board game with very simple rules. It is perfect for the minimax algorithm - there's enough complexity in minimax to use a fair amount of a language, but it can be written in a few hours if so motivated.
Written by Alex Brinkman who lives and works in Denver, but plays in the mountains.