The absence of a closed-form expression for iterated powers

Let $V$ be the vector space on which the binary operation $*$ is defined, and let $a\in V$. For positive integers $n$, define

$$a^n:=\underbrace{a*a*\dots*a}_{n\text{ times}}.$$

No-closed-form-expression hypothesis. There is no general closed-form expression for the components of $a^n$ in terms of a fixed polynomial formula with finitely many terms whose structure does not depend on $n$. The hypothesis reflects the observed combinatorial growth in the number of monomials under iteration, although the supplied text gives no proof or resolution.