For example, one’s initial self-importance (3)
I believe the answer is:
ego
'self-importance' is the definition.
(I know that self-importance can be written as ego)
'for example one's initial' is the wordplay.
'for example' becomes 'eg' (short for 'exempli gratia', 'for example' in Latin).
'initial' says to take the initial letters.
The first letter of 'ones' is 'o'.
'eg'+'o'='EGO'
(Other definitions for ego that I've seen before include "One's sense of self-importance" , "Part of the mind" , "Sense of self, conceit" , "Self-confidence" , "One's self-image" .)