  Let $p$ be an odd prime number. We study growth patterns associated with
finitely ramified Galois groups considered over the various number fields
varying in a $\mathbb{Z}_p$-tower. These Galois groups can be considered as
non-commutative analogues of ray class groups. For certain
$\mathbb{Z}_p$-extensions in which a given prime above $p$ is completely split,
we prove precise asymptotic lower bounds. Our investigations are motivated by
the classical results of Iwasawa, who showed that there are growth patterns for
$p$-primary class numbers of the number fields in a $\mathbb{Z}_p$-tower.
