Let $V = \mathbb{C}^n$ be the standard representation of $\text{GL}_n(\mathbb{C})$, and let $f(n)$ be the largest value of $k$ such that every irreducible subrepresentation $W$ of $V^{\otimes k}$ remains irreducible after restriction to some finite subgroup of $\text{GL}_n(\mathbb{C})$. What's known about the function $f$? In particular, is it unbounded?

For example, I think (but am not sure) that $f(2) = 4$, where the finite subgroup is the binary icosahedral group.