The Martin Gardner doesn't really seem to fit, since it is trivial to solve using brute force. Even if using a computer is considered cheating, there are shortcuts to use to keep from having to try everything. For example, having a "1" digit gets you nowhere, a "0" kills you, and a "5" and any even digit also kills you (and the 5 will persist at the end if you don't have an even digit, so it will probably kill you next round).

This led me to wonder, does persistence ever max out? It seems likely to me that it does.