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 lateral moves, for the bottom diagram.


![Untwisted double of a Goeritzd unknot][1]


  [1]: https://i.sstatic.net/55SHg.jpg