I recall a discussion on this question on math.sci.research quite a few years ago. John Conway had tried various moves to supplement Reidemeister moves. If I recall correctly, his guess was that no finite collection of moves will give monotonic simplification.
The Goeritz unknot below admits a simplifying move along dashed the arc shown. If one takes its untwisted double (as Ryan suggested) one again gets an unknot. (I hope I got the twists right below, but if not then adding twists shouldn't hurt). I don't see any obvious simplifying moves, or even any helpful lateral moves, for the bottom diagram.
