Not quite what you are after but the best that is currently available is the "countries #" under the statistics tab. Number of countries cached in for each individual.
In terms of how many is possible thats a tricky one. There are several places in Europe where multiple countries borders are next to each other. Including some of the really tiny countries. Note many of these operate under international agreements so there aren't border checks.
For instance France, Belgium, Luxemborg, Germany share a border within 30 miles of each other. ie: within 30 miles you could do 4 countries. The Netherlands is then only 90 miles away from them. So 5 in a day would be very simple.
Switzerland is then about 150 miles from the German/French/Luxemborg border, Lichtenstien is further 250 miles away and Austria only another 5 miles from entering the tiny country of Lichtenstien. Italy is then 100 odd miles south. Note though that Switzerland/Lichtenstien, Austria, Italy is travelling through the Alps mountain range so jouney times are longer, and Switzerland does have border checks.
It may therefore be possible to do 9 in a day without too much difficulty, best done in summer of course to maximise daylight and minimise snow disruption in mountains.
Note if Monaco is treated as a country you could grab this one too as its about 150 miles from entering Italy.
Hope this helps.
Edit: Just checked Monaco does count. Also if you really want to push it you could skip over to San Marino (300 miles from Monaco) and Vatican City (160 miles from San Marino) both within Italy to make it 12 in a day.
