Celebrity making one gawp endlessly (4)
I believe the answer is:
star
'celebrity' is the definition.
(I know that star is a type of celebrity)
'gawp endlessly' is the wordplay.
'gawp' becomes 'stare' (I've seen this before).
'endlessly' means to remove the last letter.
'stare' with its last letter taken away is 'STAR'.
'making one' is the link.
(Other definitions for star that I've seen before include "Big name" , "Pointed symbol" , "Eg, Betelgeuse" , "Ball of fire" , "It twinkles" .)