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).
Lyndon B. Johnson
As he had served less than 24 months of President Kennedy's term, Johnson was constitutionally permitted to run for a second full term in the 1968 presidential election under the provisions of the 22nd Amendment.
Initially, no prominent Democratic candidate was prepared to run against a sitting president of the Democratic Party. Only Senator Eugene McCarthy of Minnesota challenged Johnson as an anti-war candidate in the New Hampshire primary, hoping to pressure the Democrats to oppose the Vietnam War. On March 12, McCarthy won 42 percent of the primary vote to Johnson's 49 percent, an amazingly strong showing for such a challenger. Four days later, Senator Robert F. Kennedy of New York entered the race. Internal polling by Johnson's campaign in Wisconsin, the next state to hold a primary election, showed the President trailing badly. Johnson did not leave the White House to campaign.