During the NHL season, the team plays up to a hundred times. These can be daily games, or with a break of several days. You will find an overview of the current and recent season's schedule of matches, teams playing too frequency.
So, 3 matches in 3 days:
Season 2019-2020. There were too many such cases in a full season, so that are the teams, played 3 times during 3 days, several times:
Season 2018-2019, 6 times has New York Islanders, and too many teams has 5 to list it here...
As for 4 matches in 4 days, there have been no such cases in the current season.
Season 2019-2020 (4 games in 4 days):
Same for season 2018-2019:
The following teams had to play 5 times in 6 days: