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), I do not allow integrals or derivatives, I do not allow substitutions into functions with multiple variables etc. Also $n! = \prod_{i=1}^{n} i$ doesn't qualify since that formula is not built from binary operations.

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.

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), 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). I am also interested in solutions that use real or complex numbers but remember, no taking integer parts!

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.

How to prove that there is no formula for $n!$ that does only use the binary operations $+,-,*,/$ and powers on 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), I do not allow integrals or derivatives, I do not allow substitutions into functions with multiple variables etc. Also $n! = \prod_{i=1}^{n} i$ doesn't qualify since that formula is not built from binary operations.

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.

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), I do not allow integrals or derivatives, I do not allow substitutions into functions with multiple variables etc. Also $n! = \prod_{i=1}^{n} i$ doesn't qualify since that formula is not built from binary operations.

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.