In any of the games in the Super Smash Bros. series, the combo counter refers to a function in Training mode designed to keep track of combos. For each consecutive hit on a character, the combo counter will rise by 1, until the combo ends or until the player faults and allows for the hitstun to elapse. The combo counter will register a hit in the combo if it was made before more than one frame of non-hitstun. If not, the counter resets. Grabs and landing or teching will reset the combo counter.
In Super Smash Bros. Melee, the maximum number of consecutive hits that the game can record is 32,767. This irregular value can most likely be explained by binary data storage, as it's equal to 215 – 1.