Downright eccentric to go topless (5)
I believe the answer is:
utter
'downright' is the definition.
(I've seen this before)
'eccentric to go topless' is the wordplay.
'eccentric' becomes 'nutter' (I've seen this before).
'to go topless' means to remove the first letter.
'nutter' with its initial letter taken away is 'UTTER'.
(Other definitions for utter that I've seen before include "Make sounds (but not necessarily words)" , "in toto" , "(Of counterfeit money) put into circulation - say" , "Put into words" , "Say; total" .)