How to prove that there is no formula for $n!$ that does only use the binary operations $+,-,*,/$ on natural numbers, powers of natural numbers, and fixed natural numbers?

Similar question have been asked several times on math.SE but none of the answers there provide a formula of the specified type.

I am quite sure that there is no such formula but I have no clue how one could prove that.

In particular, I do not allow integer parts (floors or ceilings), except that the division function returns the quotient, so it does hide a sort of floor operation, which is allowed in this form.

I do not allow integrals or derivatives, sum or product symbols, I do not allow substitutions into functions with multiple variables etc. The length of the formula should be uniformly bounded for all $n$ (otherwise $1\cdot2\cdot\ldots\cdot n$ would work).

My motivation is that my 9-year-old son asked me this question. This doesn't mean that your solution should be understandable by a 9-year-old, but rather that I only allow the operations that he has heard of. Also note that most likely there is no such formula but I want a proof for that.