  In this article we consider the maximum possible growth rate of sequences of
long products of $d \times d$ matrices all of which are drawn from some
specified compact set which has been normalised so as to have joint spectral
radius equal to $1$. We define the marginal instability rate sequence
associated to such a set to be the sequence of real numbers whose $n^{th}$
entry is the norm of the largest product of length $n$, and study the general
properties of sequences of this form. We describe how new marginal instability
rate sequences can be constructed from old ones, extend an earlier example of
Protasov and Jungers to obtain marginal instability rate sequences whose limit
superior rate of growth matches various non-integer powers of $n$, and
investigate the relationship between marginal instability rate sequences
arising from finite sets of matrices and those arising from sets of matrices
with cardinality $2$. We also give the first example of a finite set whose
marginal instability rate sequence is asymptotically similar to a polynomial
with non-integer exponent. Previous examples had this property only along a
subsequence.
