Necessary conditions for being expressible using hyper-operators

Question

We can recursively define expressions involving only addition and multiplication as follows:

1. Let $K$ be a set of constant functions, usually the integers or some field.
2. Functions in $K$ are defined to be expressible, the identity function is expressible, and if $f$ and $g$ are expressible, so are $f+g$ and $fg$.
3. Only functions obtainable by applications of the above two rules are expressible.

The set of functions we get are the polynomials. In searching for a closed form for a function, it's usually straightforward to rule out a closed form using only $+$ and $\times$ because there are several strong necessary conditions on being a polynomial:

1. The function must be eventually bounded by a polynomial.
2. The function must have a finite number of zeroes.
3. The function cannot take on a particular value an infinite number of times.
4. The functions derivatives must eventually be zero.
5. The function must be continuous and differentiable.

But suppose we want to rule out a closed form involving exponentiation as well? Or a closed form involving arbitrary hyperoperators? For simplicity, you could assume we're only talking about positive integer functions.