The most common and most likely answer to this crossword clue is the 8 letter word EGOMANIA
I've seen this crossword clue in The Times
E
G
O
M
A
N
I
A
More clues leading to the result EGOMANIA
- Fashion image on a vanity case?
- Self-obsession of Arab in revolutionary period
- In time, soon over obsessive self-love
- Self-obsession of say, king deposed from European state
- In advancing years, soon over obsessive preoccupation with me
- Obsessive self-interest, for example, East European uncovered
- Say leader deposed from European state for self-indulgence?
- Asian in time standing for self-centredness
- Abnomal self-concern, say, shown by leaderless country
- This can provide an image that includes love – for oneself?
Buddy explains!

The straight part
We know that "Self-obsession" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case self-obsession = egomania.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does EGOMANIA mean?
EGOMANIA noun- Excessive vanity, pride or arrogance; self-importance.