Greek islander having no right to catch fabulous sea creature (8)
I believe the answer is:
cetacean
'sea creature' is the definition.
'greek islander having no right to catch fabulous' is the wordplay.
'greek islander' becomes 'cretan' (I've seen this before).
'having no' is a deletion indicator.
'right' becomes 'r' (common abbreviation).
'to catch' is an insertion indicator.
'fabulous' becomes 'ace' (both can mean excellent).
'cretan' with 'r' removed is 'cetan'.
'cetan' going around 'ace' is 'CETACEAN'.
(Other definitions for cetacean that I've seen before include "Generic name for whale, dolphin or porpoise" , "Large marine mammal" , "Member of the animal order containing whales and dolphins" , "One swimming" , "Animal order including whales and dolphins" .)