  Let K be a number field, t a parameter, F=K(t) and f in K[x] a polynomial of
degree d. The polynomial P_n(x,t)= f^n(x) - t in F[x] where f^n is the n-fold
iterate of f, is absolutely irreducible over F; we compute a recursion for its
discriminant. Let L=L(f) be the field obtained by adjoining to F all roots, in
a fixed algebraic closure, of P_n for all n; its Galois group Gal(L/F) is the
iterated monodromy group of f. The iterated extension L/F is finitely ramified
if and only if f is post-critically finite (pcf). We show that, moreover, for
pcf polynomials f, every specialization of L/F at t=t_0 in K is finitely
ramified over K, pointing to the possibility of studying Galois groups with
restricted ramification via tree representations associated to iterated
monodromy groups of pcf polynomials. We discuss the wildness of ramification in
some of these representations, describe prime decomposition in terms of certain
finite graphs, and also give some examples of monogene number fields.
