Lots of superior chaps keeping private (7)
I believe the answer is:
umpteen
'lots of' is the definition.
(I've seen this before)
'superior chaps keeping private' is the wordplay.
'superior' becomes 'u' (I've seen this in other clues).
'chaps' becomes 'men' (I've seen this before).
'keeping' means one lot of letters goes inside another.
'private' becomes 'pte'.
'men' enclosing 'pte' is 'mpteen'.
'u'+'mpteen'='UMPTEEN'
(Other definitions for umpteen that I've seen before include "Quite a lot of (sl)" , "Indefinitely large number" , "Indefinitely many" , "Ever so many" , "Informally, very many" .)