Almost set out to be a celebrity (4)
I believe the answer is:
star
'a celebrity' is the definition.
(I know that star is a type of celebrity)
'almost set out' is the wordplay.
'almost' means to remove the last letter (most of the word but not all of it).
'set out' becomes 'start' (synonyms).
'start' with its final letter taken away is 'STAR'.
'to be' is the link.
(Other definitions for star that I've seen before include "For example, Marilyn Monroe" , "Heavenly apparition" , "Cult sci-fi film" , "Five-pointed figure" , "- - of Bethlehem" .)