There are several Chess rules related to this question.
First, no number of consecutive checks automatically creates a checkmate.
Second, there is only one limit to the length of a game without a checkmate, called the 50-move rule. You could put your opponent in check 49 times without a checkmate, but on the 50th move (assuming there has been no pawn move or capture) the game is a stalemate (a draw).
Third, many checkmates occur the very first time the enemy king is placed in check.
Fourth, there is a rule called "perpetual check". If you can show that no matter what your opponent does, you can again put him in check every move, you can call a stalemate (draw). This is a useful rule if you have a big disadvantage but the enemy king is exposed.
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1, a checkmate ends the game. However, there can be an infinite number of checks.