A U.S. president can only serve two full terms in office. The limiting of terms is found in the 22nd Amendment to the Constitution, which says that "No person shall be elected to the office of the President more than twice."
Prior to the ratification of the 22nd Amendment in 1951, only one president, Franklin Roosevelt, was elected more than twice. He was elected four times. The most any president can serve under the stipulations of this amendment is 10 years. A vice-president who fills the remaining time in a president's unfulfilled term up to two years is still eligible to run for two of his own terms. If the unfinished term is more than two years, the vice-president can only run for one term.