The most common and most likely answer to this crossword clue is the 10 letter word EGOMANIACS
I've seen this crossword clue in The Times
E
G
O
M
A
N
I
A
C
S
Buddy explains!
The straight part
We know that "Fans of number one" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case fans of number one = egomaniacs.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 EGOMANIACS mean?
EGOMANIACS noun- A person obsessed with their own (supposed) importance.