I have been on a quest to create a visualization of the entire US judicial system.
So far, I've gathered enough information to do this for the federal courts, and the state appellate courts. This includes about 400 courts. Here's a screenshot of that (the interactive visualization is available at https://googledrive.com/host/0B2GQktu-wcTiWm82NGt5MTZreHM/):
There are lots of other courts within the states, of course, and diving into that is next on my list.
I came across the National Center for State Courts web site that has, among other things, a count of the number of courts within the states. This includes circuit courts, district courts, municipal courts, probate courts, and even "Town and Village Justice" courts in New York. I am not sure exactly how accurate these numbers are, but it provides some perspective as to how big the total number is.
The figure below provides some insight into how much farther I've got to go with this visualization:
My current visualization of the US judicial system
includes only the courts in that little green wedge.
"Interactive State Court Organization App",
available at http://www.ncsc.org/sco# )
Whatever the precise numbers are, it is clear that there's a ways to go here.