8/8/7K/p7/R3b2k/6rP/4p3/4B1N1 w - - 0 1
This is an age old record by Shinkman: Black is stalemated if having the move, but the White move 1.Bb4 grants 32 moves. (Don't bother trying to top that...)
Now Black doesn't have to be stalemated, and two variants suggest themselves. Can you beat my attempts?
a) Black to move would have x moves. But White is to move. After White's move Black has y moves. Maximize y-x. (Possibly replacing the B with a Q is already the answer.)
b) White to move has x moves. After White's move, if White would be to move again, White would have y moves. Maximize y-x. (I already have a position which probably yields the record.)