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
Also typically permitted is for one person to seek multiple offices at the same level of government in the same election, although attempting to simultaneously seek multiple offices in the same branch of government (e.g. a sitting U.S. Representative seeking re-election to the House "and" election to the U.S. Senate) is severely frowned on and prohibited in many states (the constitutionality of these prohibitions is uncertain). Recent examples include three 2000 and 2012 presdential elections where Senator Joe Lieberman and Representative Paul Ryan respectively sought re-election and election to the vice presidency - neither was elected vice president, but both were re-elected to the offices in which they were the incumbents.