Offensively curious or inquisitive (5)
I believe the answer is:
nosey
This clue is a double definition.
'offensively curious' is the first definition.
(I've seen this in another clue)
'inquisitive' is the second definition.
(I've seen this in another clue)
(Other definitions for nosey that I've seen before include "Curious in nasal way" , "Offensively curious or inquisitive" , "Impertinent" , "Showing too much interest in other people's affairs" , "Prying" .)