Can we characterize those non-constant polynomials $f(x) \in \mathbb Z[x]$ such that the set $\{|f(n)| : n \in \mathbb Z\}$ contains infinitely many "highly composite" (https://en.wikipedia.org/wiki/Highly_composite_number) numbers ?
Also see this related Asymptotic for the number of distinct prime factors of the non-zero values of non-constant polynomials