Completely happy English composer? Almost completely (8)
I believe the answer is:
blissful
'completely happy' is the definition.
(blissful means very happy)
'english composer? almost completely' is the wordplay.
'english composer?' becomes 'bliss' (Arthur Bliss, f89f-f975).
'almost completely' becomes 'ful' (I can't justify this - if you can you should give a lot more credence to this answer).
'bliss'+'ful'='BLISSFUL'
(Other definitions for blissful that I've seen before include "Joyous, rapt" , "Completely happy and contented" , "Delighted" , "Happy, often delusively" , "Full of joy" .)