Technically, if a VP serves two years or more as president after taking over the office, they can only run for more term after that.

TIL (I'm not an American).

Welcome to the wonders of American Politics. Twenty-second amendment of the constitution prohibits it:

No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once.

This was put into place after Franklin Delano Roosevelt was elected to a fourth term as President following WW2, in 1945. Prior to that there was no limit on the number of terms a person could serve as President, because the founding fathers presumed that someone being a career politician would never be a thing. Up until that point it had been tradition that the sitting president would only remain for two terms, if elected, and then bow out to allow for new blood. FDR died in office, Truman took over, and in 1951 the amendment was ratified as a direct response to Roosevelt's four terms. Sadly, they really should have put in term limits for the congresscritters as well, because there's way too many of them that are still sitting that have been there through multiple presidents.