The most common and most likely answer to this crossword clue is the 7 letter word FIXATED
I've seen this crossword clue in The Daily Telegraph
F
I
X
A
T
E
D
Buddy explains!

The straight part
We know that "with an obsession" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case with an obsession = fixated.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 FIXATED mean?
FIXATED verb- To make something fixed and stable; to fix.
- To stare fixedly at something.
- To attend to something to the exclusion of all others; used with on.
- To attach oneself to a person or thing in a pathological or neurotic manner; used with on.
- Attached to someone or something in a neurotic or pathological manner