  Let $m_n(G)$ denote the number of maximal subgroups of $G$ of index $n$. An
upper bound is given for the degree of maximal subgroup growth of all
polycyclic metabelian groups $G$ (i.e., for $\limsup \frac{\log m_n(G)}{\log
n}$, the degree of polynomial growth of $m_n(G)$). A condition is given for
when this upper bound is attained.
  For $G = \mathbb{Z}^k \rtimes \mathbb{Z}$, where $A \in GL(k,\mathbb{Z})$, it
is shown that $m_n(G)$ grows like a polynomial of degree equal to the number of
blocks in the rational canonical form of $A$. The leading term of this
polynomial is the number of distinct roots (in $\mathbb{C}$) of the
characteristic polynomial of the smallest block.
