What nice properties does exponentiation have?

Exponentiation of course satisfies a number of nontrivial identities:

However, these identities all involve functions other than exponentiation (I’m thinking of $0$ and $1$ as nullary functions, here). My question is what identities hold of exponentiation alone. That is:

What is the equational theory of $(\mathbb{N}, exp)$?

To be clear, I mean “identity” in the strict, universal-algebraic sense: one term equals another term, where each term is built from variables and exponentiation alone. Also, an identity has to hold on all of $\mathbb{N}$: identities which hold only on, say, numbers divisible by $17$ don’t count.

A related question:

Is that theory axiomatized by finitely many equations?

Note: A previous version of this question asked whether there were any nontrivial identities at all. This was extremely silly of me, as pointed out almost immediately by Stefan Perko below: $(x^y)^z=(x^z)^y$.

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