Large primate but not quite top (3)
I believe the answer is:
ape
'large primate' is the definition.
(I've seen this before)
'not quite top' is the wordplay.
'not quite' means to remove the last letter.
'top' becomes 'apex' (I've seen this before).
'apex' with its final letter taken away is 'APE'.
'but' acts as a link.
(Other definitions for ape that I've seen before include "Imitate like primate" , "Large primate" , "drill" , "Take off" , "gorilla" .)