A white king on g6, a black king on h8, nothing else anywhere. Try to win it. The white king cannot go to g7 or h7, because the black king covers both. The black king cannot go to them either. He steps to g8, back to h8, and nothing in the position ever changes.
8 
A lone king cannot even give check
Check means the enemy king is attacked. A king attacks the eight squares around it and nothing else, so to attack the black king it would have to be standing next to him, and a king may never move onto a square an enemy piece covers. Every square next to a king is covered by that king. The check never happens, and with no check there is no mate. It is the same rule that stops a king taking a king.
I put this through a computer once because a student would not believe me: on an empty board with only the two kings there is no legal position at all where the player to move has no move. Two bare kings cannot even stalemate each other. Whoever is being chased always has somewhere to go.
The game is over before you get to try
Strip everything else off and the game does not politely continue until somebody blunders. King against king is a dead position: no series of legal moves by either player can produce checkmate, so it is drawn the instant it arises, and neither player claims anything. Article 5.2 of the FIDE Laws covers it, and it is the simplest case of insufficient material.
The clock will not rescue you either. If your opponent’s flag falls while you have nothing but a king, you do not win on time. The game is a draw, because you could not have mated him by any possible series of moves (Art. 6.9). I have watched a junior run his opponent down to zero with a bare king, look up in triumph, and get half a point.
One rook changes everything
8 
d The rook gives the check. The king takes the flight squares. Neither piece could do the job alone, and that division of labour is every basic mate there is: king and queen against king, king and rook, two bishops. Your king is not a spectator once the queens are off. It is the second attacker, and players who leave it sitting on g1 while the queen tours the board are the ones who take twenty-five moves over a ten-move job.
So no, not with a bare king. And not without your king either: all a king contributes to a mate is squares taken away, but no mate works without it. If you are the one down to nothing, a lone king can never win and the game is drawn already, so stop sweating. If your opponent is, get the finish clean on mate-in-one puzzles before you next need it with a clock running.