Weary old American wearing hat (7)
I believe the answer is:
exhaust
'weary' is the definition.
(exhausting is a kind of wearying)
'old american wearing hat' is the wordplay.
'old' becomes 'ex' (prefix meaning former or onetime).
'american' becomes 'US' (United States).
'wearing' is an insertion indicator.
'us' inserted into 'hat' is 'haust'.
'ex'+'haust'='EXHAUST'
(Other definitions for exhaust that I've seen before include "get everything out of" , "lower part of estate?" , "Completely tire out" , "Weary" , "Empty" .)