  Let $\mathcal{A}$ be a finite set of $d\times d$ matrices with integer
entries and let $m_n(\mathcal{A})$ be the maximum norm of a product of $n$
elements of $\mathcal{A}$. In this paper, we classify gaps in the growth of
$m_n(\mathcal{A})$; specifically, we prove that $\lim_{n\to\infty} \log
m_n(\mathcal{A})/\log n\in\mathbb{Z}_{\geqslant 0}\cup\{\infty\}.$ This has
applications to the growth of regular sequences as defined by Allouche and
Shallit.
