Using my copy of the Thomas Cook European Rail Timetable, we came to the conclusion that one could leave northeastern Italy and cross Austria, Liechtenstein, Switzerland, Germany, France, Luxembourg, and Belgium, finally arriving in Maastricht, Netherlands, all within 24 hours. There was enough slack in the schedule we came up with that it might even be possible to start in Yugoslavia, if the connections were with you.
As it turns out, someone did it, more or less the way we thought:
According to http://www.britishcouncil.org/learnenglish-central-trivia-trains.htm:
"The greatest number of countries travelled through entirely y [sic] train in 24 hours is 10, by Aaron Kitchen on 16-17 February 1987. His route started in Yugoslavia and continued through Austria, Italy, Liechtenstein, Switzerland, France, Luxembourg, Belgium and the Netherlands, arriving in Germany 22 hours and 42 minutes later."
We also discussed whether it'd be possible to transit the same nine countries by car in the same period.
We never did get a chance to try, even after he got an old beater of a French car. But now, many years later with the help of Google Maps I have an answer .
It turns out that you can do nine countries easily, as you can cover a lot more ground driving. In fact, in 22 hours and 33 minutes, Google Maps says you can do this:
Starting in Lanžhot in the Czech Republic you drive though Slovakia, Hungary, Croatia, Slovenia, Italy, Austria, Liechtenstein, Switzerland, Germany, France, Luxembourg, and Belgium, finally crossing the border into the Netherlands just about twenty-two and a half hours later. That's fourteen countries.
(Google Maps fails to do general directions in Bosnia-Hercegovina, Serbia, or Romania; they will only give you point-to-point between major cities, apparently. When they do, I think it will be possible to change the route so that it transits those countries instead of the Czech Republic and Slovakia, and thus gain another country, for a total of fifteen countries in twenty-four hours.)
Of course, this assumes zero time for food, toilet stops, and fuel. On the other hand, it also assumes you don't exceed the speed limit. Since the above route has a fairly minimal distance in Germany, where there is no speed limit--what does Google Maps use as an assumed speed on the unlimited bits of the autobahn, anyway? The advisory limit of 130km/h?--it all depends on how strict the speed enforcement is.
By the way, that Google Maps route suggests one might beat an earlier road record from 1977 I found online:
177. Most countries in 24 hours
Ralph C. Johnson, Dean Rittenhouse and Herman Oldigs drove a hired Mercedes 280S 2,032km/1,262.6 miles in 23hr 33min on 11 May 1977, visiting 11 European countries.
Of course, there are more countries now than there were before the Iron Curtain fell and Czechoslovakia, the Soviet Union, and Yugoslavia broke up, so it's really comparing two different situations. But that just makes for more possible routes.
I'm trying ViaMichelin as well, but I'm finding their interface makes this many stopovers difficult.