  We first show that a class of pro-$p$ branch groups including the Grigorchuk
group and the Gupta-Sidki groups all have subgroup growth type $n^{\log n}$. We
then introduce the notion of orbit growth and use it to construct extensions of
the Grigorchuk group and the Gupta-Sidki groups. We compute the subgroup growth
type of these extensions and deduce that all functions between $n^{(\log n)^2}$
and $e^n$ occur as the subgroup growth type of a pro-$p$ group, thus, giving
almost complete answer to a question raised by Lubotzky and Segal.
