  Given a field $K$, a rational function $\phi \in K(x)$, and a point $b \in
\mathbb{P}^1(K)$, we study the extension $K(\phi^{-\infty}(b))$ generated by
the union over $n$ of all solutions to $\phi^n(x) = b$, where $\phi^n$ is the
$n$th iterate of $\phi$. We ask when a finite extension of
$K(\phi^{-\infty}(b))$ can contain all $m$-power roots of unity for some $m
\geq 2$, and prove that several families of rational functions do so. A
motivating application is to understand the higher ramification filtration when
$K$ is a finite extension of $\mathbb{Q}_p$ and $p$ divides the degree of
$\phi$, especially when $\phi$ is post-critically finite (PCF). We show that
all higher ramification groups are infinite for new families of iterated
extensions, for example those given by bicritical rational functions with
periodic critical points. We also give new examples of iterated extensions with
subextensions satisfying an even stronger ramification-theoretic condition
called arithmetic profiniteness. We conjecture that every iterated extension
arising from a PCF map should have a subextension with this stronger property,
which would give a dynamical analogue of Sen's theorem for PCF maps.
