Bill's thinking (9)
I believe the answer is:
reckoning
I believe this is a double definition.
'bill's' is the first definition.
(reckoning is a kind of bill)
'thinking' is the second definition.
(reckoning is a kind of thinking)
(Other definitions for reckoning that I've seen before include "Settling of accounts" , "estimating?" , "Count" , "Counting, calculating" .)