It is a term that is prepended to an adjective to avoid using a superlative. It was used old and is now little used (only by grandparents). You can replace the quantity adverbs quite, too much or too much.
'Asaz' is an almost disused word, still seen in poetry and literature. It means "something, quite, not too much, enough". It comes to us from the occitan assatz, which is a variation of Latin ad satis ( "to satiety" ).
asaz: high or high degree of an expression against a position, or a situation.