Differences between polynomials and factorials

Let $A_n$, $n=1,2,3,\dots$, be an increasing sequence of positive integers such that for each prime number $p$, there is an integer $s=s(p)$ for which $p|A_n$ for every $n\geq s$. Such are, for instance, the sequences of factorials $A_n=n!$, least common multiples of the first $n$ positive integers $A_n=\operatorname{LCM}(1,2,\dots,n)$, and the products of the first $n$ primes $A_n=p_1 p_2\cdots p_n$. For any of such sequences $A_n$, $n=1,2,3,\dots$, and any polynomial $f\in\mathbb{Z}[x]$ of degree at least $2$, we show that the set of positive integers that are not expressible as $f(x)-A_n$ for some $x,n\in\mathbb{N}$ has a positive lower density. We also investigate the case when $f\in\mathbb{Z}[x]$ is linear and a few related problems. It is shown, for instance, that $x^2-ay^2=n!$ has infinitely many integer solutions in $(x,y,n)\in\mathbb{N}^3$, where $n\geq 2$, for each integer $a$ in the range $1\leq a\leq 12$, but has no such solution for $a=13$.