Cheap gifts only
This one may seem a little bit odd, but this rule is in place to make sure that no one can bribe the President easily with expensive gifts. If a gift given to the President is worth more than three hundred and seventy-five dollars, it is immediately given to the national archive.
The President is not allowed to keep it, just everything else that is less expensive can have and be classified as a souvenir or mark of courtesy. We guess this helps them from being corrupted by lavish gifts, but it must suck when you just want to receive that new Bugatti.