Drink after drink — going topless, very laid back (6)
'very laid back' is the definition.
(supine means lying on one's back)
'drink after drink going topless' is the wordplay.
I cannot really see how this works, but
'drink' could be 'sup' (to sup is to drink) and 'sup' is found within the answer.
This may be the basis of clue (or it may be nonsense).
(Other definitions for supine that I've seen before include "Lying face upward" , "Inert; on one's back" , "Unresisting" , "Slow in acting" , "Lying face upwards, indolent" .)
'drink' is the definition.
(I've seen this before)
'drink going topless very laid back' is the wordplay.
'drink' becomes 'cha' (cha can mean tea).
'going topless' means to remove the first letter.
'very' becomes 'tres' ('very' in French).
'laid back' says the letters should be written backwards.
'tres' with its initial letter removed is 'res'.
'res' in reverse letter order is 'ser'.
'after' is the link.
(Other definitions for chaser that I've seen before include "Beer after spirits e.g." , "One who pursues - one drink after another" , "Pursuit cannon; one engraving" , "Horse; drink following another" , "one in the national hunt" .)