  Let $K$ be a number field with ring of integers $\mathcal O_K$, and let
$\{f_k\}_{k\in \mathbb N}\subseteq \mathcal O_K[x]$ be a sequence of monic
polynomials such that for every $n\in \mathbb N$, the composition
$f^{(n)}=f_1\circ f_2\circ\ldots\circ f_n$ is irreducible. In this paper we
show that if the size of the Galois group of $f^{(n)}$ is large enough (in a
precise sense) as a function of $n$, then the set of primes $\mathfrak
p\subseteq\mathcal O_K$ such that every $f^{(n)}$ is irreducible modulo
$\mathfrak p$ has density zero. Moreover, we prove that the subset of
polynomial sequences such that the Galois group of $f^{(n)}$ is large enough
has density 1, in an appropriate sense, within the set of all polynomial
sequences.
