An "Amazon" (R+B+N) starting in one of the 4 corner squares of the board can mate an enemy king that is placed on one of the 4 center squares in 4 moves all by itself.
A piece used in "Grand Chess" called the Cardinal (B+N) can mate an enemy king all by itself on an otherwise empty board but the king has to cooperate by stupidly moving to a corner square.
However, the Grand Chess piece called the Marshall (R+N) cannot mate by itself, interestingly enough, and needs other pieces to help.