As a numbers/statistics geek I would like to be able to look up information on how "busy" particular airports or regions are.
I'm sure the FAA collects this data, but is it accessible to the general public?
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The FAA's Air Traffic Activity Data System (ATADS) contains broad statistical information on activity in the national airspace system which can be used to get an idea of how "busy" a particular Center, TRACON, or Tower facility is.
The caveat is this data only encompasses aircraft talking to Air Traffic Control, so it won't be able to tell you anything about uncontrolled fields except possibly what you can glean from Center/TRACON activity information for the greater area.
I'm not sure where they get their data from, but Fltplan.com has annual reported data for each airport in their Airports & FBO's database:
For example, this is for KBRY (which is uncontrolled):