  In this paper, we investigate the existence of multiple positive solutions to
the following multi-critical Schr\"{o}dinger equation \begin{equation}
\label{p} \begin{cases}
  -\Delta u+\lambda V(x)u=\mu
|u|^{p-2}u+\sum\limits_{i=1}^{k}(|x|^{-(N-\alpha_i)}*
|u|^{2^*_i})|u|^{2^*_i-2}u\quad \text{in}\ \mathbb{R}^N,\\ \qquad\qquad\qquad
u\,\in H^1(\mathbb{R}^N), \end{cases} \end{equation} where $\lambda,\mu\in
\mathbb{R}^+, \, N\geqslant 4$, and $2^*_i=\frac{N+\alpha_i}{N-2}$ with
$N-4<\alpha_i<N,\,i=1,2,\cdots,k$ are critical exponents and
$2<p<2^*_{min}=\min\{2^*_i:i=1,2,\cdots,k\}$. Suppose that
$\Omega=int\,V^{-1}(0)\subset\mathbb{R}^N$ is a bounded domain, we show that
for $\lambda$ large, problem above possesses at least $cat_\Omega(\Omega)$
positive solutions.
