Given a field $K$ that is neither algebraically closed nor real closed, must there necessarily be an irreducible polynomial of degree $n$ over $K$ for all positive integers $n$?
By the Artin-Schreier theorem, we know that irreducible polynomials of arbitrary large degrees must exist, but this does not necessarily mean that such polynomials exist for all degrees.
No. Let $K=\bigcup_n{\mathbb{F}_{p^{3^n}}}$. Then any polynomial in $K$ of degree $3^d >1$ is reducible, but the algebraic closure of $K$ is of infinite degree over $K$.
The same could be said by replacing the sequence $(3^n)_n$ with any increasing sequence $a_n$ such that $a_n|a_{n+1}$ and such that there exists a prime $q$ such that $v_q(a_n)$ is bounded: then there will be irreducible polynomials of degree a power of $q$ over $K$, but no irreducible polynomials of degree $r$ if $v_r(a_n)$ diverges.