Given a field [Formula: see text] and [Formula: see text], we say that a polynomial [Formula: see text] has newly reducible [Formula: see text]th iterate over [Formula: see text] if [Formula: see text] is irreducible over [Formula: see text], but [Formula: see text] is not (here [Formula: see text] denotes the [Formula: see text]th iterate of [Formula: see text]). We pose the problem of characterizing, for given [Formula: see text], fields [Formula: see text] such that there exists [Formula: see text] of degree [Formula: see text] with newly reducible [Formula: see text]th iterate, and the similar problem for fields admitting infinitely many such [Formula: see text]. We give results in the cases [Formula: see text] as well as for [Formula: see text] when [Formula: see text]. In particular, we show that for all these [Formula: see text] pairs, there are infinitely many monic [Formula: see text] of degree [Formula: see text] with newly reducible [Formula: see text]th iterate over [Formula: see text]. Curiously, the minimal polynomial [Formula: see text] of the golden ratio is one example of [Formula: see text] with newly reducible third iterate; very few other examples have small coefficients. Our investigations prompt a number of conjectures and open questions.