In a reachable game, what is the maximum sum of queens, rooks, bishops, knights, and kings on the board?
For queens only, see Maximum number of queens possible
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@Aschultz is correct that 26 pieces is the maximum number of non-pawn and non-king pieces on the board. Since they said that they have not seen such proof yet, here it is to be seen.
[Title "me, chess.stackexchange.com 6/18/2019, Non-Unique Proof Game In 40.5 Moves"] [FEN ""] 1. d4 a5 2. b4 c5 3. h4 e5 4. f4 g5 5. dxc5 axb4 6. hxg5 exf4 7. g4 b5 8. a4 h5 9. c4 f5 10. c6 f3 11. c5 f4 12. g6 b3 13. g7 b2 14. Na3 Nh6 15. g5 b4 16. g6 b3 17. g8=B b1=B 18. Bh7 Ba2 19. g7 b2 20. g8=B b1=B 21. e4 d5 22. e5 d4 23. e6 d3 24. Kd2 Ke7 25. Kc3 Kf6 26. e7 d2 27. Qc2 d1=B 28. e8=B f2 29. c7 f3 30. c6 Bb7 31. Bg2 f1=B 32. c8=B Bfe2 33. Bcd7 f2 34. c7 f1=B 35. c8=B h4 36. a5 Ra6 37. Rh3 Rb6 38. Rg3 h3 39. a6 h2 40. a7 h1=B 41. a8=B
I haven't seen a proof that the maximum number of non-pawns is 26. It seems intuitive, but let's try and nail it down.
In the starting position, no pawns can be promoted without captures.
So the question is, what is the maximum amount of promotions one capture can create? Say a pawn goes from column x to column y.
So each pawn capture means a maximum of 3 pawns that can promote.
But there is a special case! What if there are doubled pawns in 2/3? They are dealt with as follows: if a previous capture moved them there, then it created fewer than 3 passed pawns, because they were not passed before this move. In the case of 2, the capturing x-pawn blocked them. In the case of 3, the opposing y-pawn blocked them.
So each capture means at most 3 pawns can promote. 1 means 3 promotions, 2 means 6, 3 means 9, and 4 means 12. But that would leave no further pawns to promote.
Now, to find a maximal case, we can just follow the link in the game, or play your game out so we promote all pieces to bishops. I have to admit I didn't fully check your game, but the idea seems right.
[FEN ""] 1.g4 h5 2.g5 Nh6 3.gxh6 g5 4.h7 g4 5.b4 Rg8 6.h8=B g3 7.b5 a5 8.f4 e5 9.f5 Na6 10.bxa6 Rb8 11.a7 b5 12.a8=B b4 13.f6 Be7 14.fxe7 f5 15.c4 Kf7 16.e8=N f4 17.c5 d5 18.e4 d4 19.Na3 b3 20.c6 Bd7 21.cxd7 Qc8 22.d8=B c5 23.Nb5 c4 24.Ba3 b2 25.Bc5 c3 26.Ba3 b1=B 27.Bb4 c2 28.Ba3 c1=B 29.Bb4 f3 30.Bd3 f2+ 31.Ke2 f1=B+ 32.Ke1 g2 33.Nh3 g1=B 34.Ba3 Qd7 35.Bb4 Qd5 36.Ba3 Kg6 37.exd5+ Kh6 38.d6 Rb7 39.d7 Rb8 40.Bb6 Rb7 41.d8=B e4 42.Qa4 e3 43.Qc4 Bg2 44.Qb3 Bc6 45.Kf1 Bf2 46.Bc4 Bg3 47.Kg1 e2 48.Bd3 e1=B