Horror story German playwright hasn't begun (7)
I believe the answer is:
chiller
'horror story' is the definition.
'german playwright hasn't begun' is the wordplay.
'german playwright' becomes 'schiller' (Friedrich Schiller, 1759-1805).
'hasn't begun' means to remove the first letter.
'schiller' with its initial letter taken off is 'CHILLER'.
(Other definitions for chiller that I've seen before include "Cooling device" , "Cold cabinet" , "Refrigerator" , "Scary book" , "part of fridge" .)