Twenty-second Amendment to the United States Constitution
Some argue that the 22nd Amendment and 12th Amendment bar any two-term president from later serving as vice president as well as from succeeding to the presidency from any point in the presidential line of succession. Others contend that the original intent of the 12th Amendment concerns qualification for "service" (age, residence, and citizenship), while the 22nd Amendment, concerns qualifications for "election", and thus (strictly applying the text) a former two-term president is still eligible to "serve" as vice president (neither amendment restricts the number of times an individual can be elected to the vice presidency), and then succeed to the presidency to serve out the balance of the term (though prohibited from running for election to an additional term).
Vice President of the United States
The term of office for both the vice president and the president is four years. While the Twenty-Second Amendment generally restricts the president to two terms, there is no such limitation on the office of vice president, meaning an eligible person could hold the office as long as voters continued to vote for electors who in turn would reelect the person to the office; one could even serve under different presidents. This has happened twice: George Clinton (1805–1812) served under both Thomas Jefferson and James Madison; and John C. Calhoun (1825–1832) served under John Quincy Adams and Andrew Jackson. Additionally, neither the Constitution's eligibility provisions nor the Twenty-second Amendment's presidential term limit explicitly disqualify a twice-elected president from serving as vice president. As of the 2016 election cycle however, no former president has tested the amendment's legal restrictions or meaning by running for the vice presidency.