Wonder what distinction you make between classic and mythological; they kinda overlap, don't they?
I like monsters that can be used for non-combat roles as well. Take a dragon, it may be a foe (even, yuck, a random monster), but it may be a creature you will seek for other reasons than combat and plunder. For example, you may try to negociate to enlist its help in an ongoing conflict, or to learn forgotten lore that only such an intelligent and long-lived creature might still know.
This, together with its powerful symbolism and its universality, is what makes dragons so interesting.