Number of Maximum Overlap in n-Dimensions Given a list of compact axis-aligned intervals (in 1-D), rectangles (in 2-D), cuboids (3-D) etc, what is the maximum number that overlap at any point?

In 1-D there's a fairly simple solution that exploits the the ordering of R: assuming you have a set of pairs $\{[a_1,b_1],...,[a_n,b_n]\}$ sort all of the a's and b's (together) then scan from a_min to b_max adding one to the count of overlap whenever you encounter $a_i$ for any i and subtracting one whenever you see a $b_j$ for any j. Then just keep track of the maximum value of count that you see.

I'm not sure how to extend this to 2-D, let alone n-D. Any ideas?
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.