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).
Dual mandate
Unlike many federations, U.S. states do not generally restrict state or federal officials from seeking office at another level of government without resigning their existing offices first. For example, in the four U.S. presidential elections contested from 1988 to 2000 inclusive, three sitting state governors were nominated for the presidency, these being Michael Dukakis in 1988, Bill Clinton in 1992 and George W. Bush in 2000 (Clinton and Bush were elected president), while in 2016 sitting governor Mike Pence was elected vice president. Elsewhere, serving state officials often seek federal office, one prominent example being Illinois State Senator Barack Obama's election to the United States Senate in 2004 - Obama quickly resigned from the Illinois Senate after being elected to the U.S. Senate despite not being legally required to do so, and served as a U.S. senator until 2008 when he was elected president. Also, it is not uncommon for sitting federal officials to contest election to state offices, although in these cases the office sought is usually one of the state's highest political posts, typically governor - one such recent example being the aforementioned Mike Pence who was a sitting U.S. Representative when he was first elected governor.