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Is there an integer polynomial $f\in \mathbb{Z}[x]$ such that for infinitely many prime numbers $p$ we have that

$\forall k\in\mathbb{N}: p\not | f(k)$?

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    If $p\equiv 3\pmod 4$, then $p\nmid k^2+1$ for all $k$? – 2017-02-28
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    All non-linear irreducible polynomials satisfy your condition. If $f$ is irreducible of degree $>1$, then there's a set of primes of positive density such that $f(x)$ has no root mod $p$. This is a classical consequence of Frobenius Density Theorem, see p. 48 of this thesis for instance: http://www.mat.uniroma3.it/scuola_orientamento/alumni/laureati/pesiri/sintesi.pdf – 2017-02-28

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Yes, for example $f(x)=x^2+1$. The corresponding primes $p$ are primes of the form $4k+3$.