  Let $\mathbb F_q$ be the finite field with $q$ elements, $f, g\in \mathbb
F_q[x]$ be polynomials of degree at least one. This paper deals with the
asymptotic growth of certain arithmetic functions associated to the
factorization of the iterated polynomials $f(g^{(n)}(x))$ over $\mathbb F_q$,
such as the largest degree of an irreducible factor and the number of
irreducible factors. In particular, we provide significant improvements on the
results of D. G\'{o}mez-P\'{e}rez, A. Ostafe and I. Shparlinski (2014).
