Arcade game supplying a great time after a lot of beer (7)
I believe the answer is:
pinball
'arcade game' is the definition.
(I've seen this before)
'a great time after a lot of beer' is the wordplay.
'a great time' becomes 'ball' (I can't justify this - if you can you should believe this answer much more).
'after' says to put letters next to each other.
'a lot of' means to remove the last letter (most of the word but not all of it).
'beer' becomes 'pint'.
'pint' with its last letter taken off is 'pin'.
'ball' put after 'pin' is 'PINBALL'.
'supplying' acts as a link.
(Other definitions for pinball that I've seen before include "It may require tilting when played" , "Game involving an enclosed table, played by operating flippers" , "Slot-machine game" , "Electrically operated form of bagatelle" , "Slot machine bagatelle" .)