The only Vice Presidents to have served under more than one Presidents are: George Clinton and John C. Calhoun. George Clinton replaced Aaron Burr as Thomas Jefferson's running mate; where he became VP soon afterward, next he served as James Madison's VP very soon after that.
The same can be said with John C. Calhoun who, in the 1824 election decided to run as Vice President, winning and serving under John Quincy Adams. Then he served under Andrew Jackson, however their relationship wasn't all that great and Martin Van Buren soon replaced him.
I don't know if it's possible today, with term limits and what not. It would be cool to happen again.
Vice Presidents aren't affected by term limits. So far there have been only 6 VP Presidents to serve a full 8 years, though. 4 more served over 7 years; 2 had shortened terms, 1 died, and 1 resigned. Biden is currently the 11th longest serving.
