It is possible in a chess game to have a dead position in which both players have all eight pawns, one bishop, and a king--20 pieces total. If the two players' pawns are interlocked in "zig-zag" fashion and each player's bishop is the same color as his pawns, each player's army will be forever stuck behind his own wall of pawns, with no way to ever reach anything on the other side.
[FEN "4kb2/8/1p1p5/pPpPp1p1/P1P1PpPp/5P1P/8/4KB2 w - - 0 1"]
What is the largest number of pieces that can be on the board in a legally-reachable position such that either:
The side on move has at least one legal move, but no sequence of legal moves would produce checkmate.
It would be possible to play an arbitrary number of legal moves, but no sequence of legal moves would produce checkmate.
No legal sequence of legal moves would produce checkmate or stalemate.
I would guess that the first of those might be possible with as many as 28 pieces on the board, but most escape-proof positions rely upon immobilized kings to block opposing pawns, making it hard to allow anything other than pawns to move safely. What are the actual limits?