The most common and most likely answer to this crossword clue is the 9 plus 4 letter word ONE-TRACK MIND
I've seen this crossword clue in Financial Times
O
N
E
-
T
R
A
C
K
M
I
N
D
Buddy explains!
The straight part
We know that "Obsession" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case obsession = one-track mind.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.