I'm looking for a function $P(x)=a_1 x^{b_1}+a_2 x^{b_2}+\ldots+1$ such that for any positive integer $x$ input, there is guaranteed to be a prime factor $p>x$ which divides $P(x)$. For instance, although $f(x):=x^2+1$ is not such a polynomial, it usually has the behavior I'm looking for where e.g. $5^2+1=2 \cdot 13$ and $13>5$.
This is trivially easy to do with factorials or primorials, of course. Unless I am mistaken this is also the case for $2^n-1$ and similar exponential expressions. Presumably that includes various recurrence relations built along the same mechanism.
Anyway, are there any polynomials that have this property? If not, also welcome are other function classes which do have it.