My position has a fourteenth check too, but that's probably ok. :-)
These are the moves:
And here's the whole solution uploaded to Lichess.
EDIT: Looks like 15 is also doable:
The final check feels like such a waste of a piece though; the first fourteen checks are possible with white having only two pieces and a king:
Again, here's the Lichess link.
EDIT: A winning knight underpromotion I found. All other promotions lead to wins, but underpromoting to a knight is the fastest mate (mate in 1).
EDIT: Thanks to RosieF for catching a huge error in the bishop part.
Here's a situation where promoting to queen is the only winning move:
1 h8=Q! wins
1 h8=R? only draws:
Here's a situation where promoting to bishop is best:
Promoting to bishop
Promoting to queen
Promoting to knight
Promoting to rook
As luck would have it, this morning, YouTube recommended to me a video discussing a win-study by Selezniev which ...