According to Wikipedia, termination of [Beggar My Neighbour](https://en.wikipedia.org/wiki/Beggar-my-neighbour) (a game of no skill whatsoever also known as Strip Jack Naked and less printable things) is still open and was listed by John Conway as an [anti-Hilbert problem](http://library.msri.org/books/Book42/files/guy.pdf) that should _not_ drive mathematical research. They call it long standing; I think it was in the 1970’s that I heard of it (from [Alan Ross](https://en.wikipedia.org/wiki/Alan_S._C._Ross) of U/non-U fame, oddly enough).