An interesting thing happened yesterday in a game between my son and my father: a double check, in which the moved piece was not one of the checking pieces. (In a usual double check, a piece moves, placing the king in check but also discovering a check by another piece. To quote a formulation on the U.S. Chess Federation Site, “Double check is a more dangerous form of a discovered check where not only the hidden piece attacks the king, but also the piece that moves.”) How did this happen? Everyone was following the normal rules of chess.
Note that a move that promotes a pawn, thus creating a check from the promoted piece and a discovered check because of the pawn move, would not qualify as an answer, since the moved piece (albeit not the same one that started the move) is one of the checking pieces.