score:5
I have had a lot of looking at sites like this (Traveller Century Club, Most Traveled People, etc), and while there's a US Centric one (The All Fifty States Club), I've never seen a European one. Perhaps it's time to start one? :)
Of course, the biggest problem you'll have is - what counts as a visit? This blog post has a great description of the argument faced - is it when you set foot? Leave the airport? Eat a meal? Sleep one light? What counts as visiting?
Then you have the problem of what IS a country? Is Gibraltar? What about Kosovo?
There's also an Extra Milers Club - which records visits to all 3142 counties of The US :)