Majel Barrett Roddenberry (23 February 1932 – 18 December 2008; age 76) was a recurring actress in the Star Trek franchise and was the wife of Star Trek creator Gene Roddenberry. Her most frequent portrayal in Star Trek, besides the computer, was that of Christine Chapel on Star Trek: The Original Series, Star Trek: The Animated Series, and in two of the films. She also voiced M’Ress and several other characters on The Animated Series and later played Betazoid Ambassador Lwaxana Troi on Star Trek: The Next Generation and Star Trek: Deep Space Nine. Her first filmed appearance was in the original Star Trek pilot episode “The Cage” as Number One in 1964. link
Jeffrey Alan Combs (born 9 September 1954; age 70) is an actor who has the distinction of portraying nine different characters on four Star Trek television series link
Vaughn Dale Armstrong, Jr. (born 7 July 1950; age 74) is a veteran Star Trek actor who is noted for portraying twelve separate characters in twenty-eight episodes over four separate series. link
Does the voice of the ship’s computer not count as an “appearance?”
She ranks #3, after everyone’s favorite Chief Engineer O’Brien, per IMDB.
Edit: It should be noted that after Majel’s passing and subsequent new series, Dorn dominates all.