My response was "About a dozen times", but the right answer is "I don't know". I decided to fix this and sit down with my current and expired passports and go through the stamps. Yes, it's a little tedious, but worthwhile. Here is what I got:
All the Asia travel is since 1998. I have a country breakdown for "W. Europe" when I first started traveling there in 1992, but after a few years all the entries and exits became EU, so I just put it all as W. Europe. This doesn't include Mexico, which doesn't stamp U.S. passports.
I wasn't too far off my Taiwan estimate. Japan is actually less than I thought, but maybe that is because I have spent more than two months in the country altogether.