Which country is bordered by the greatest number of other countries?
It's a close race. Russia borders on Norway, Finland, Estonia, Latvia, Lithuania (via its Kaliningrad exclave), Poland (via Kaliningrad), Belarus, Ukraine, Georgia, Azerbaycan, Kazakhstan, China, and Mongolia, and is separated by only a few kilometers of water from Japan (in the Kuriles) and the U.S. (the Diomedes in the Aleutian chain).
Update: Russia also borders on North Korea, a border so short I missed it before, and (amazingly enough) on Switzerland: there is a monument to the Russian general Alexander Suvorov near Göschenen in central Switzerland that is legally Russian territory.
China borders on North Korea, Russia, Mongolia, Kazakhstan, Kyrgyzstan, Tajikistan, Afghanistan, Pakistan, India, Nepal, Bhutan, Myanmar, Laos, and Vietnam. If you call Taiwan a country, the Taiwanese island of Jīnmén (better known as Quemoy) is also only a few kilometers from China.