score:12
I just performed some Google search that gave me the map of Europe during Napoleonic wars.
Here goes the reproduction from Wikipedia:
See the image in other resolutions here: Europe 1812 map en.png
So, in Europe, in 1812, there were the following countries:
I think doing just simple googling would help.
Please remember that Napoleonic Wars was not one war and countries (except Great Britain) did not fight in each of them.
Upvote:8
No major European country stayed neutral (for the whole time of Napoleonic wars).
The Ottoman Empire (partly placed in Europe, although not traditionally considered a European country) possibly comes closest. But it was by no means the boring Swiss-style neutrality, so I would review the Napoleon's Egyptian campaign, the Russo-Ottoman War of 1806-1812, and the 1807 British naval intervention to get the full story.
Source: http://books.google.com/books?id=cG6JC_E1Rf8C&pg=PA269 combined with http://books.google.pl/books?id=h_TguykH1qcC&pg=PA8 to rule out Kingdom of Sardinia; combined with http://books.google.pl/books?id=QjzYdCxumFcC&pg=PA223 for Ottoman Empire.
Update: Ottoman Empire was in a war with France until 1802.