Was just thinking about this. It's easy to have two pieces checking the king via a discovered double check, but three is where it gets interesting. Finegold said in a video he was TDing for a game and a player's king was in check from five different pieces. Sounds like an exaggeration, but I'm curious what input people have.
If you have an unlimited number of pieces to place, and ignore the mechanics needed to get them there in actual gameplay, it's easy to show you can surround a king with eight queens and eight knights for a total of sixteen checks.
In actual gameplay, however, you have only fifteen non-king pieces to use, and there are serious difficulties in getting more than two pieces to an attacking position in the same move, remembering that the opposing player cannot leave his king in check after his own move. Answers to this question even claim that triple check is impossible in standard chess, never mind quintuple.
Under standard rules, a double check can result from a move giving both a direct and a discovered check. In the case of an en-passant capture, it is even possible to invoke a double discovered check, but the moving pawn is then not in position to give a third direct check.
It is also possible to consider more than two pieces to be "attacking" the opposing king, in the form of batteries (which would need to be set up via discovered check) or absolute pins. However, these do not directly count as checking pieces, only as complications to the problem of getting out of check.
Do you have a link to Ben Finegold's video? He's a respected player, so I'm sure there's some context being missed here.