Curiously Romeo starts handling coins, initially from greasiest till (4,8)
I believe the answer is:
cash register
'till' is the definition.
(cash register is a kind of till)
'curiously romeo starts handling coins initially from greasiest' is the wordplay.
'curiously' indicates an anagram.
'starts' suggests taking the first letters.
'initially' says to take the initial letters.
'from' says to put letters next to each other.
The first letter of 'romeo' is 'r'.
The first letters of 'handling coins' is 'hc'.
'r'+'hc'+'greasiest'='rhcgreasiest'
'rhcgreasiest' is an anagram of 'CASH-REGISTER'.
(Other definitions for cash register that I've seen before include "Machine used in shops" , "one may advise change" , "money machine" , "Shop till" .)