The most common and most likely answer to this crossword clue is the 8 letter word SUPERFAN
I've seen this crossword clue in Independent
S
U
P
E
R
F
A
N
Buddy explains!

The straight part
We know that "Obsessive" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case obsessive = superfan.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 SUPERFAN mean?
SUPERFAN noun- An extremely dedicated fan