According to About.com :
China and Russia have the most neighboring countries with 14 neighbors each.
China borders Afghanistan, Bhutan, India, Kazakhstan, Kyrgyzstan, Laos, Mongolia, Myanmar (Burma), Nepal, North Korea, Pakistan, Russia, Tajikistan, and Vietnam.
Russia borders Azaerbaijan, Belarus, China, Estonia, Finland, Georgia, Kazakhstan, Latvia, Lithuania, Mongolia, North Korea, Norway, Poland, and Ukraine. (This list includes the two countries that border Kaliningrad - Poland and Lithuania. Otherwise, Russia has 12 neighbors.)
Reference: About.com (2016) Which country borders the most other countries? Available from < http://geography.about.com/library/faq/blqzmostneighbors.htm > [19 Feb 2016]