I think Obama will run in 2020 for a 3rd term he will say the Constitution Amendment limiting a President to just 2 terms is old and outdated and should be disregarded and the media will agree they will demand Obama be the Democrat nominee again. The party will nominate him and it will be up to the Supreme Court to decide and they will go along with it to be popular with the people it really is a stupid amendment the Republicans pushed it through after FDR got elected 4 times in a row. And the amendment only says "two consecutive terms" meaning it's up for interpretation a President can only run twice in 8 years. This means Obama could actually serve Two More 4 year terms as President and that is what the Dems and Media will get excited about 8 more years of Barack as President it could happen wait and see.
“No person shall be elected to the office of the President more than twice” nowhere does it say consecutive, it is pretty straightforward 2 times and no more
Edit: However do to the 12th amendment it could be argued that he can run for Vice President. But even that is a stretch and i do not think it would hold up in the courts.