Intrinsically unpopular, mad, illogical creature (9)
I believe the answer is:
armadillo
'creature' is the definition.
(armadillo is a kind of creature)
'intrinsically unpopular mad illogical' is the wordplay.
'intrinsically' indicates the answer is hidden within the clue.
'ARMADILLO' is hidden in the letters of 'unpopular mad illogical'.
(Other definitions for armadillo that I've seen before include "Burrowing mammal with a covering of bony plates" , "Molar laid out for the horny-plated animal" , "Bony-plated insectivore" , "American insect-eater" , "Insectivorous mammal" .)