A U.S. president's eight-year term is it. That's been the limit, ever since FDR won four straight elections and died in office in 1945. But there is no law saying a vice president can't continue to serve in that capacity. Indeed, the VP spot is often like the batter's box -- stand in there and you're up next. (The most recent VP to become president was George H.W. Bush.)
So when wikileaks released a March email containing the names of 39 people Hillary Clinton's team was putting forward for consideration for her running mate, it's a little surprising Vice President Joe Biden wasn't on the list, especially since Biden is part of the most popular administration (according to the polls) in recent memory. Instead there were senators, cabinet secretaries, business leaders, philanthropists and governors. But no Vice President Biden. Here's the list, in the original groupings.
Claire McKaskill [sic]
Corey [sic] Booker
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