There are at least 4 games per best of seven series with a maximum of 7 games per series played. There are a total of 15 series (7 series per conference plus 1 for the Stanley Cup). That's at least 60 games with a maximum of 105 games (if every series goes to game 7).
Each team plays 4 series. That means they play at least 16 games with a maximum of 28 games (assuming the team makes it to the Stanley Cup playoffs).
TL&DR:
16-28 games per team (assuming they go to the Stanley cup Playoffs).
60-105 games overall.
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There can be a total of 56 games in the first round if they all went to 7 games.
During the play-offs, a team must win a minimum of 16 games and a maximum of 28 games to win the Stanley Cup, there are 4 round, best of 7 games.