user12377: I was looking at minimal combinations of material needed to force checkmate (K+R, Q+B, R+N+N, etc). You end up with a sort of threshold of how much material is needed, and that sort of assigns a strength to the king. But that made me wonder if K+K is sufficient material to force checkmate. Obviously this is normally impossible, but in the hypothetical endgame where white has two kings and black has one (and make the obvious loosenings of the restrictions on moving into check), could the two kings force checkmate against the one king?
Alternatively, one could think of it as white having a king and a fairy chess piece that moves exactly like a king. Is this enough material to force checkmate?

user9765: Non-royal king is a combination of [0,1] leaper (wazir) and [1,1] leaper (fers) and in fairy chess called a erlking. About erlking, see my book Fairy chess endings on an n x n chessboard (2017), page 545.
The ending of king + erlking against bare king on board 8x8 is won for the stronger side and the longest win has 18 moves. Also the ending of two erlkings against king is a general win and the longest win has 16 moves.

user3819: A King and a Man (or commoner; i.e., a non-royal King) can mate a lone King. The longest distance to mate on a standard 8x8 board is 18 moves.
The result quoted above was obtained by H.G. Muller in 2008, see this coment on chessvariants.com: http://www.chessvariants.com/index/listcomments.php?id=28770

user11629: If these are two real kings, it is not possible. This is because one requirement of checkmate is that the black king is in check which cannot be achieved with any of the white kings, since it would put the white king in check.
If you say that white has only one king and a nonstandard chess piece that moves exactly like a king (i.e. this piece can be put next to the enemy king), then yes, it seems possible and quite easy to mate the black king.
For example, put the king and “king” for on c3 and f3, and the black king on d5. As you can see the two kings almost cover the whole 4th rank. Pushing the black king to the side of the board and into the corner is quite easy, e.g. if black starts with 1... Ke5, 2. Kc4 Kd6 3. Kf4 Ke6 4. Kc5. Once you have the black king around the corner you put white's real king on c6 and give checkmate with the "king", the piece that moves like a king but really isn’t one, on g7. 
Here is a demonstration of said strategy in the replayer. Remember, white’s standard king is on b2 and their other “king” is on f3!
[FEN "8/8/8/3k4/8/5K2/1K6/8 w - - 0 1"]

1. Kc3 Ke5 2. Kc4 Kd6 3. Kf4 Ke6 4. Kc5 Kd7 5. Kf5 Ke7 6. Kc6 Ke8 7. Kf6 Kd8 8. Kf7 Kc8 9. Ke8 Kb8 10. Kd8 Ka8 11. Kb6 Kb8 12. Kdc7+ Ka8 13. Kcb7#


12377: <p>I was looking at minimal combinations of material needed to force checkmate (K+R, Q+B, R+N+N, etc). You end up with a sort of threshold of how much material is needed, and that sort of assigns a strength to the king. But that made me wonder if K+K is sufficient material to force checkmate. Obviously this is normally impossible, but in the hypothetical endgame where white has two kings and black has one (and make the obvious loosenings of the restrictions on moving into check), could the two kings force checkmate against the one king?</p>

<p>Alternatively, one could think of it as white having a king and a fairy chess piece that moves exactly like a king. Is this enough material to force checkmate?</p>
 9765: <p>Non-royal king is a combination of [0,1] leaper (wazir) and [1,1] leaper (fers) and in fairy chess called a erlking. About erlking, see my book <a href="http://problem64.beda.cz/silo/kotesovec_endings_on_an_nxn_chessboard_2ed_2017.pdf" rel="nofollow noreferrer">Fairy chess endings on an n x n chessboard</a> (2017), page 545.</p>
<p>The ending of king + erlking against bare king on board 8x8 is won for the stronger side and the longest win has 18 moves. Also the ending of two erlkings against king is a general win and the longest win has 16 moves.</p>
 3819: <p>A King and a Man (or commoner; i.e., a non-royal King) can mate a lone King. The longest distance to mate on a standard 8x8 board is 18 moves.</p>

<p>The result quoted above was obtained by H.G. Muller in 2008, see this coment on chessvariants.com: <a href="http://www.chessvariants.com/index/listcomments.php?id=28770" rel="noreferrer">http://www.chessvariants.com/index/listcomments.php?id=28770</a></p>
 11629: <p>If these are two real kings, it is not possible. This is because one requirement of checkmate is that the black king is in check which cannot be achieved with any of the white kings, since it would put the white king in check.</p>

<p>If you say that white has only one king and a nonstandard chess piece that moves exactly like a king (i.e. this piece can be put next to the enemy king), then yes, it seems possible and quite easy to mate the black king.</p>

<p>For example, put the king and “king” for on c3 and f3, and the black king on d5. As you can see the two kings almost cover the whole 4th rank. Pushing the black king to the side of the board and into the corner is quite easy, e.g. if black starts with 1... Ke5, 2. Kc4 Kd6 3. Kf4 Ke6 4. Kc5. Once you have the black king around the corner you put white's real king on c6 and give checkmate with the "king", the piece that moves like a king but really isn’t one, on g7. </p>

<p>Here is a demonstration of said strategy in the replayer. Remember, white’s standard king is on b2 and their other “king” is on f3!</p>

<pre><code>[FEN "8/8/8/3k4/8/5K2/1K6/8 w - - 0 1"]

1. Kc3 Ke5 2. Kc4 Kd6 3. Kf4 Ke6 4. Kc5 Kd7 5. Kf5 Ke7 6. Kc6 Ke8 7. Kf6 Kd8 8. Kf7 Kc8 9. Ke8 Kb8 10. Kd8 Ka8 11. Kb6 Kb8 12. Kdc7+ Ka8 13. Kcb7#
</code></pre>
