Anyone at first, latest in long sequence sick in crowded tenement (7)
I believe the answer is:
anthill
'crowded tenement' is the definition.
(anthill can mean a crowded building)
'anyone at first latest in long sequence sick' is the wordplay.
'at first' says to take the initial letters.
'latest in long sequence' becomes 'nth' (nth can denote the latest in a sequence).
'sick' becomes 'ill' (similar in meaning).
The first letter of 'anyone' is 'a'.
'a'+'nth'+'ill'='ANTHILL'
'in' is the link.
(Other definitions for anthill that I've seen before include "Nest" , "Built-up colony" , "Mound of insects" , "Mountain (of insects)" , "Home for social workers?" .)