A braggart has a big one (3)
I believe the answer is:
ego
'a braggart has a big one' is the definition.
(I've seen this in another clue)
This is all the clue.
(Other definitions for ego that I've seen before include "number one" , "Self-confidence" , "Self-esteem or self-importance" , "Sense of self, conceit" , "One's sense of self-importance" .)