The most common and most likely answer to this crossword clue is the 9 letter word ONE-TRACK
I've seen this crossword clue in The Times
O
N
E
-
T
R
A
C
K
More clues leading to the result ONE-TRACK
Buddy explains!

The straight part
We know that "Obsessed with" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case obsessed with = one-track.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.