Complexity of Finding Largest Set of Intersecting Convex Polytopes
Suppose that the dimension $d$ of the Euclidean space is fixed, and that the input consists of $n$ convex polytopes in $\mathbb{R}^d$ that altogether have $p$ facets.

Let $h_1,\ldots,h_p$ denote the supporting hyperplanes for the $p$ facets of the input polytopes. The arrangement of $h_1,\ldots,h_p$ is the decomposition of $\mathbb{R}^d$ into connected open cells of dimensions $0,1,\ldots,d$. 
It is well-known that the arrangement has combinatorial complexity $O(p^d)$ and that it can be computed in time $O(p^d)$; see for instance


  Chapter 28 "Arrangements" by D. Halperin and M. Sharir,
  in Handbook of Discrete and Computational Geometry (edited by J.E. Goodman, J. O'Rourke, and C.D. Tóth)
  https://www.csun.edu/~ctoth/Handbook/chap28.pdf


So you may compute the arrangement in polynomial time $O(p^d)$, and then compute for every cell $C$ in the arrangement the number $x(C)$ of input polytopes that contain $C$.
The maximum of all these numbers $x(C)$ yields the answer to your question.