With the "Use as many and any black or white pieces (both kings included)" restriction, the theoretical maximum becomes

61 checks and captures: there must be a square for both of the kings, and a target square for white's first move. The rest can be full of capturable white pieces.

Obviously, there can be no legal position that can have that many white pieces, so we throw legality out of the window and start from here:

[FEN "PPPPPPPP/PPPPPPPP/PPPPPPPP/PPPPPPPP/PPPPPPPP/QPPPPPPP/QQPPPP1P/KQQQQRPk w - - 0 1"]

And as long as we are careful to eat the bottom two pawn rows in good order, we are good to go for the maximum:

For some strange reason, Lichess declines to do anything with this position, and chess.com won't let me copy paste the moves, so you get a screenshot of the board from one, and a screenshot of the moves from another. If you have a tool that works better, please drop a comment.

With the "Use as many and any black or white pieces (both kings included)" restriction, the theoretical maximum becomes

61 checks and captures: there must be a square for both of the kings, and a target square for white's first move. The rest can be full of capturable white pieces.

Obviously, there can't be any legal position with that many white pieces, so we throw legality out of the window and start from here:

[FEN "PPPPPPPP/PPPPPPPP/PPPPPPPP/PPPPPPPP/PPPPPPPP/QPPPPPPP/QQPPPP1P/KQQQQRPk w - - 0 1"]

And as long as we are careful to eat the bottom two pawn rows in good order, we are good to go for the maximum:

For some strange reason, Lichess declines to do anything with this position, and chess.com won't let me copy paste the moves, so you get a screenshot of the board from one, and a screenshot of the moves from another. If you have a tool that works better, please drop a comment.