Renowned American doctor beset by endless bother (6)
I believe the answer is:
famous
'renowned' is the definition.
(synonyms)
'american doctor beset by endless bother' is the wordplay.
'american doctor' becomes 'amo' (I can't explain this - if you can you should give a lot more credence to this answer).
'beset by' means one lot of letters goes inside another ('beset' could historically mean 'cover').
'endless' means to remove the last letter.
'bother' becomes 'fuss' (synonyms).
'fuss' with its final letter taken away is 'fus'.
'amo' inserted within 'fus' is 'FAMOUS'.
Can you help me to learn more?
(Other definitions for famous that I've seen before include "Well-known" , "Known to very many people" , "Renowned, celebrated" , "Renowned, noted" , "Known and esteemed" .)