The others taking a new king to be most offensive (7)
I believe the answer is:
rankest
'most offensive' is the definition.
('rank' can be similar in meaning to 'offensive')
'the others taking a new king' is the wordplay.
'the others' becomes 'rest' (I've seen this before).
'taking' means one lot of letters goes inside another.
'new' becomes 'n' (common abbreviation eg NT for New Testament).
'king' becomes 'K' (abbreviation for king in chess).
'a'+'n'+'k'='ank'
'rest' enclosing 'ank' is 'RANKEST'.
'to be' acts as a link.
(Other definitions for rankest that I've seen before include "most offensive" , "most objectionable" .)