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).
2020 United States Senate elections
One-term incumbent Democrat Cory Booker was re-elected in 2014; Booker first took office after winning a 2013 special election. Booker, who is running for President in 2020, could run for both President and re-election to the Senate as New Jersey election laws allow candidates to simultaneously run for two offices. Should Booker choose to forgo a Senate run in 2020, though, potential Democratic nominees include incumbent governor Phil Murphy and incumbent Congressmen Donald Norcross and Josh Gottheimer.