  The joint spectral radius of a bounded set of $d \times d$ real matrices is
defined to be the maximum possible exponential growth rate of products of
matrices drawn from that set. For a fixed set of matrices, a sequence of
matrices drawn from that set is called \emph{extremal} if the associated
sequence of partial products achieves this maximal rate of growth. An
influential conjecture of J. Lagarias and Y. Wang asked whether every finite
set of matrices admits an extremal sequence which is periodic. This is
equivalent to the assertion that every finite set of matrices admits an
extremal sequence with bounded subword complexity. Counterexamples were
subsequently constructed which have the property that every extremal sequence
has at least linear subword complexity. In this paper we extend this result to
show that for each integer $p \geq 1$, there exists a pair of square matrices
of dimension $2^p(2^{p+1}-1)$ for which every extremal sequence has subword
complexity at least $2^{-p^2}n^p$.
