The most common and most likely answer to this crossword clue is the 9 letter word MONOMANIA
I've seen this crossword clue in The Daily Telegraph
M
O
N
O
M
A
N
I
A
More clues leading to the result MONOMANIA
- Girl involves Arab in obsession
- Single-minded obsession of woman nursing Arab
- Record set by fellow, one with a very high level of self-regard
- What’s obsessive fad for pre-stereo sound?
- Old woman's guarded on subject of sultan's obsession
- Obsession with single old record
- Arab stopping Mona’s obsessive single-mindedness
- Old record collector’s obsession?
- Sun-follower, old fellow, one having a fixation
- Moan perversely about Gulf inhabitant’s obsessive single-mindedness
Buddy explains!

The straight part
We know that "obsessive single-mindedness" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case obsessive single-mindedness = monomania.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 MONOMANIA mean?
MONOMANIA noun- Excessive interest or concentration on a singular object or subject.
- A pathological obsession with one person, thing or idea.