Are there local maxima in the number of moves required to solve a Rubik's Cube?
Asking Tomas Rokicki this question immediately yielded the correct answer ("yes, local maxima exist"):


  If a position exhibits total
  symmetry, it is of necessity a local
  maximum (all except the start).  A
  little thought should make it clear
  why this is the case in the QTM [quarter-turn metric].  For
  the HTM [half-turn metric] it's a bit more subtle but not
  too bad.
  
  ...
  
  Such a position is pons asinorum,
  which is distance 12 in QTM and
  distance 6 in HTM (U2D2F2B2L2R2).


I don't see why this is the case for the half-turn metric; but for
the quarter-turn metric it is clear.  In a position with total symmetry, all neighboring positions must be at the same path length (since all moves are equivalent by symmetry).  So a position with total symmetry must be either a local maximum or a strict local minimum.  But strict local minima cannot exist... there has to be some move that reduces the distance to the solved state, just by the definition of the distance.  The symmetry argument translates to the $n \times n \times n$ cube, as does the example position provided.