Greek god giving that girl endless chaos (6)
I believe the answer is:
hermes
'greek god' is the definition.
(Hermes is an example)
'giving that girl endless chaos' is the wordplay.
'giving' is a charade indicator (letters next to each other).
'that girl' becomes 'her' (**).
'endless' means to remove the last letter.
'chaos' becomes 'mess' (I've seen this before).
'mess' with its last letter removed is 'mes'.
'her' put next to 'mes' is 'HERMES'.
(Other definitions for hermes that I've seen before include "Winged messenger" , "Greek god of commerce and theft" , "Greek messenger of the gods, like Roman Mercury" , "Greek messenger god" , "Messenger and herald of the gods" .)