  We consider a $p$-fractional Choquard-type equation \[ (-\Delta)_p^s
u+a|u|^{p-2}u=b(K\ast F(u))F'(u)+\varepsilon_g |u|^{p_g-2}u \quad\text{in
$\mathbb{R}^N$}, \] where $0<s<1<p<p_g\leq p_s^*$, $N \geq \max\{2ps+\alpha,p^2
s\}$, $a,b,\varepsilon_g\in (0,\infty)$, $K(x)= |x|^{-(N-\alpha)}$, $\alpha\in
(0,N)$, and $F$ is a doubly critical nonlinearity in the sense of the
Hardy-Littlewood-Sobolev inequality. It is noteworthy that the local
nonlinearity may also have critical growth. Combining Brezis-Nirenberg's method
with some new ideas, we obtain the ground state solutions via the mountain pass
lemma and a generalized Lions-type theorem.
