The most common and most likely answer to this crossword clue is the 11 letter word ILLUSTRIOUS
I've seen this crossword clue in Financial Times
I
L
L
U
S
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Buddy explains!

The straight part
We know that "Famous" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case famous = illustrious.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 ILLUSTRIOUS mean?
ILLUSTRIOUS adjective- Admired, distinguished, respected, or well-known, especially due to past achievements or noble qualities.