Now how about in a position in which there aren’t any promoted pieces?
I’m not sure if that would be more interesting or less. Of course, an upper bound would be 27 (for the queen) plus 28 (14 for each rook) plus 26 (13 for each bishop) plus 16 (8 for each knight) plus 32 (4 for each pawn) plus 8 (for the king). If I’ve added correctly that’s 137.
Bill Smythe
Probably time for a topic split. Four for each pawn assumes moving one or two squares on the first move or having a choice of two captures, but the a and h pawns would only have the opportunity for one capture and any capture opportunity would eliminate the possibility of forward moves of the pawn in front of the piece that could be captured. Pawns on a2, c2, e2. h2, b4, d4, f4 and g6 would have as many as 26 moves (with enemy pieces on b3, d3, f3, g3, a5, c5, e5, g5, f7, h7). Maximizing the pawn moves really cuts down on the mobility of the other pieces.
If the a- and h-pawns each moved 2 squares and then captured on b5 and g5 respectively, then all 8 pawns would have 2 capture opportunities and 2 non-capture opportunities. But it would be impossible to enable all 32 of those opportunities. For example, if black pieces are placed on c3 and e3 to enable d-pawn captures, then the c- and e-pawns would lose non-capture opportunities.
Furthermore, increasing one pawn’s non-captures tends to cut down on other pawns’ captures.
Bill Smythe
I don’t understand how a pawn on b5 has 2 non-capture opportunities regardless of what else is on the board. I can see how a white pawn on b7 can have as many as twelve moves.
Alex Relyea
Oops, you’re right. 
I never count multiple promotions on the same square (and from the same square) as 4 possible moves, only 1.
Bill Smythe
I’ve moved the last several replies here from A Rating Related Question in Running Chess Tournaments.