Foreword: In my humble opinion, promoting illegally technically counts as not promoting in a sense, as in it is not anorthodox, legal promotion.
I feel that a list of all possible possible illegal promotions should be made. As such, I shall do so.
Here is a list of all that I can think of and what I have come across. Even though some examples have already been shown in other answers, they must be used as the only known example to me. I will use different examples if I can.
Promoting Into A:
- A king
- A pawn
- Keeping it as a pawn, but the promoting some moves later
- An enemy king
- An enemy pawn
- An enemy queen
- An enemy rook
- An enemy bishop
- An enemy knight
I will provide examples of each, assume that it is White to play and win with a one exception.
You can find a few more of these funny problems here: https://www.huffpost.com/entry/chess-solutions-weird-pro_b_680681
[Title "Charles Henry Wheeler, 1132 Dubuque Chess Journal 07/1875, Mate In Two"]
[FEN "5R2/3P4/4k3/8/3K4/8/8/8 w - - 0 1"]
1. d8=WK Kd6 2. Rf6#
An enemy king
[Title "Karl Arthur Leonid Kubbel, 1941, Mate In Two"]
[FEN "8/4P1Q1/2P5/8/8/6Pk/5P2/7K w - - 0 1"]
1. e8=BK Kd8 2. Qd7#
Both of the above scenarios are combined in this wacky puzzle from this ChessBase article.
[Title "Yuri Dorogov’s Story"]
[FEN "8/P7/BB5P/8/5NN1/2K5/p6P/k7 w - - 0 1"]
“An alien came to Earth, learning chess. His teacher Yuri Dorogov won a lot of material and the diagrammed position emerged. Dorogov decided to mate in two with 1.Kc2, but having not taught the promotion rules thoroughly backfired now, as the cunning alien replied with 1...h1=K!
Indeed, now 2.Bd4 is mate for one king but stalemate for the other, so it is only a draw. Dorogov, however, exploited his mistake to his advantage and played the incredible move 2.a8=bK (see diagram), moving his pawn to a8 and promoting into a black king!
The alien had only one move 2...Kb8 and Dorogov won after all with 3.h7 Ka8 4.h8Q###, a triple checkmate.“
Additionally, here is a case that considers castling rights. So longs as the pieces have never moved and are opposite their starting squares, castling is allowed, as far as I can tell.
[Title "Andrew Buchanan,PDB Website 15/1/2019, After Anonymous Mate In 3 Without Queening A Pawn"]
[FEN "8/P1k1P3/P4P2/3P4/3B4/8/8/8 w - - 0 1"]
1. e8=K! Kc7 2. a8=R! Kd6 3. 0-0-0#!
This reminds me of Pam-Krabbe-Rochade castling. So I made a problem that uses it, but on the opposite side of the board, as illegal promotions allow for it to happen that way.
[Title "WTM, Helpmate In 2 Without Queening A Pawn"]
[FEN "1b6/1Pk1P3/1pN5/1P6/8/8/4p3/8 w - - 0 1"]
1. e8=K e1=WR 2. 0-0-0-0-0-0#
[Title "Samuel Lloyd, American Chess Magazine. December 1898, Mate In 3"]
[FEN "N1Br4/2Pb1P2/3k4/1P2R3/1P3K2/B7/8/8 w - - 0 1"]
1. cxd8=P If: 1.cxd8Q/R?-stalemate, 1.cxd8B?Bf5+!, 1.cxd8S? Bc6+!
An enemy pawn AND rook
[FEN "rn6/kPR5/8/K7/8/8/8/8 w - - 0 1"]
Mutual promotion into a pawn!
[Title "Black To Move White And Win, Geir Sune Tallaksen Østmoe After Peter Wong, Matplus, 11/12/17"]
[FEN "5qnk/3pPB2/1p1p1N2/1pbK1NP1/1p1pP3/1P1P4/6p1/7R b - - 0 1"]
1... gxh1=P! 2. exf8=P! and White easily wins with an extra piece.
Unique Promotion Into An Enemy Pawn
[Title "Mate In 3, Made By Me"]
[FEN "rk1K4/p6P/P7/2P5/8/8/8/8 w - - 0 1"]
1. h8=BP h6 (Pawns are allowed to do a double step on their first move) 2. c6 h5 3. c7#
Unique Promotion Into An Enemy Rook
[Title "Mate In 4, Made By Me"]
[FEN "1K1k1bq1/3BpPp1/4P1Pp/7P/8/6N1/8/8 w - - 0 1"]
1. fxg8=BR Rh8 2. Ne5 Rg8 3. Nc5 Rh8 4. Nb7#
Keeping it as a pawn, but the promoting some moves later
[Title "Author??? Mate In 3"]
[FEN "7K/5P2/6pp/6pk/6pr/6pb/8/8 w - - 0 1"]
1. g8 Bf1 2, f8=Q or 1... g2 f8=N
An enemy queen-Many thanks to Rosie. F and Noam Elkies for this amazing problem! Creating a problem where the only way to WIN is to promote to a Black queen is surely the holy grail of illegal pawn promotion problems!
[Title "Matt Bengtson, Chess Braintwisters (Burt Hochberg), no. 103. White to move & draw"]
[FEN "4kn2/3p1pPp/4pPpK/6P1/8/2p5/1b6/8 w - - 0 1"]
1. g8=BQ Qg7+ 2. fxg7 c2 3. g8=Q c1=Q 4. Qxf7+ Kd8 5. Qe8+ Kc7 6. Qc8+ Kb6 7. Qc6+ Ka5 8. Qb5+ Kxb5
An enemy bishop
[Title "Camille Moriau, 97 150 Chess Curiosities ,1910, Capriccio "]
[FEN "7B/5pPk/5K2/6R1/8/8/8/8 w - - 0 1"]
1. g8=BB Kh7 2. Rh5# or 1... Bh7 Bg7#
An enemy knight
[FEN "r7/kPR5/8/K7/8/8/8/8 w - - 0 1"]