  We investigate $k$-superirreducible polynomials, by which we mean irreducible
polynomials that remain irreducible under any polynomial substitution of
positive degree at most $k$. Let $\mathbb F$ be a finite field of
characteristic $p$. We show that no $2$-superirreducible polynomials exist in
$\mathbb F[t]$ when $p=2$ and that no such polynomials of odd degree exist when
$p$ is odd. We address the remaining case in which $p$ is odd and the
polynomials have even degree by giving an explicit formula for the number of
monic 2-superirreducible polynomials having even degree $d$. This formula is
analogous to that given by Gauss for the number of monic irreducible
polynomials of given degree over a finite field. We discuss the associated
asymptotic behaviour when either the degree of the polynomial or the size of
the finite field tends to infinity.
