Following is a question spun off from a comment I received:
is a factorial an elementary function and an algebraic function?
From elementary functions by Wikipedia
By starting with the field of rational functions, two special types of transcendental extensions (the logarithm and the exponential) can be added to the field building a tower containing elementary functions.
So isn't a factorial a multiplication of finite polynomials, and therefore a polynomial, a rational function, an algebraic function, and an elementary function?
Added: Now I realized a factorial cannot be a polynomial, for that it doesn't make sense to talk about its degree while it does for a polynomial