Personalities, for example, outsize (4)
I believe the answer is:
egos
'personalities' is the definition.
(ego can mean someone's personality or self)
'example outsize' is the wordplay.
'example' becomes 'eg' (short for 'exempli gratia', 'for example' in Latin).
'outsize' becomes 'os'.
'eg'+'os'='EGOS'
'for' is the link.
(Other definitions for egos that I've seen before include "Self-images" , "Lots of self-esteem" , "People's personal pride - the parts of them that are conscious" , "feelings of self-importance" .)