It follows from the prime number theorem and the periodicity properties $f(n+p) \equiv f(n) \mod{p}$ that for each $A < e$ there are only finitely many integer polynomials $f \in \mathbb{Z}[x]$ such that $|f(n)| < A^n$ for all $n \in \mathbb{N}$. On the other hand, for each $k \in \mathbb{N}$ the binomial coefficient $\binom{n}{k}$ is an integer-valued polynomial in $n$ bounded by $2^n$.

Question 1. Are there infinitely many integer polynomials with $|f(n)| < e^n$ for all $n \in \mathbb{N}$?

Question 2. Given $A < 2$, are there only finitely many integer-valued polynomials $f \in \mathbb{Q}[x]$ with $|f(n)| < A^n$ for all $n \in \mathbb{N}$?

An update on question 1. David Speyer's solution below raises the following

Follow-up question. What is the supremum of those $c \in \mathbb{R}^{> 0}$ such that, for any $t_0 < \infty$, there are only finitely many $f \in \mathbb{Z}[x]$ with $\inf_{t > t_0} |f(t)|/\Gamma(t) \leq c$?

He shows that any $c < 1/2$ has this property. On the other hand, the example of $k!\binom{x}{k}$ shows that this no longer holds with $c = 1$, and the supremum lies in the interval $[1/2,1]$.

(My apology for modifying the original Question 1: it is only in view of the proof below that I think the follow-up question might be interesting. As far as I can see, it does appear to be easy to construct an explicit sequence of examples with $c < 1$ and a fixed $t_0$.)