Number of Maximum Overlap in n-Dimensions
This is a well-known variant of Klee's measure problem.  The best algorithm known, due to Timothy Chan, runs in roughly $O(n^{d/2})$ time, ignoring polylogarithmic factors.  Faster algorithms are known for the special cases of fat boxes and boxes with one or more fixed coordinates; see recent work by Karl Bringmann and Hakan Yıldız, respectively.