Most drunk end with most sleep (8)
I believe the answer is:
tipsiest
'most drunk' is the definition.
'end with most sleep' is the wordplay.
'end with' becomes 'tip' (tip is a kind of end. I am not sure about the 'with' bit.**).
'most' means to remove the last letter.
'sleep' becomes 'siesta' (I've seen this before).
'siesta' with its final letter taken away is 'siest'.
'tip'+'siest'='TIPSIEST'
(Other definitions for tipsiest that I've seen before include "Most tired and emotional" , "Most tiddly" , "Most affected by booze" , "Most mildly drunk or fuddled" , "Most tiddley" .)