Doctor, for example, beginning to swallow sediment (5)
I believe the answer is:
'sediment' is the definition.
(dregs is a kind of sediment)
'doctor for example beginning to swallow' is the wordplay.
'doctor' becomes 'dr' ('Dr' can be a synonym of 'doctor').
'for example' becomes 'eg' ('e.g.' can be a synonym of 'for example').
'beginning to' says to take the initial letters.
The initial letter of 'swallow' is 's'.
(Other definitions for dregs that I've seen before include "Most worthless parts" , "Sediment - rabble" , "Sediment at the bottom of a liquid" , "Remnant of drink" , "Grounds; refuse" .)