While we're on the topic of weaknesses, though, might as well share some more myth...
Hercules fought two very notable creatures as part of his trials - the Hydra and the Nemian Lion.
Although the hydra is already in the game, a part of the myth that isn't already a part of the game is that the hydra had poisonous blood and heads that would regrow as he cut them off. Hercules "failed" the trial of the hydra because he had to get advice for how to defeat the hydra (cauterizing the wound with fire to prevent the head from regrowing), but after defeating the hydra, he put his arrows in the blood of the hydra, and used those as poison arrows in his subsequent adventures.
There was also the Nemian Lion, whose hide was essentially impervious to harm. Basically similar to having adamantine skin, the lion could not be harmed by conventional kinetic attack... through its hide. Some tellings of the myth state the weakness of the lion was to attack its open mouth, and crack the skull from the inside, where its hide did not protect it. (Which sounds a lot like a video game boss... "Who would have thought that the boss's only weakness was its giant, glowing red eye?") Hercules then hollowed the lion out, and used its hide as a cloak as a sort of armor.
Also, since I was trying to think of "alternate" type of monster, Celtic folktales have the Leanan Sidhe (or some variant spelling thereof). Sidhe are a general class of fairies in Celtic myth, but the Leanen Sidhe takes the form of beautiful women who act like greek muses - inspiring artists and craftsmen, but at the same time, draining away portions of their life as they pour it into their work.
Some versions of the myth have them feeding off of artistry, and encourage artists to literally put their souls onto their canvas, so that they can feed off them. Other versions have them as something like a succubus. Other versions just play the whole thing as a metaphor for romance in general, and say that any woman can take the aspect of a Leanan Sidhe, and if your love is unrequited, you become her slave, and if you resist her, and her love is unrequited, she becomes your slave.
And of course, there are the potentially beneficial kind of monster, like the Russian Domovoi, or its Japanese counterpart, the Zashiki-Warashi. These are creatures that like to live in people's houses, but hide. As long as they are living with you, they bring your family good fortune, but if they ever leave, it will fall immediately to ruin. As such, if you have one of those, you have to find ways to ensure it can't ever leave, even if that means trying to find a way to cage it. Domovoi are generally entirely benign, unless angered, although Zashiki-Warashi are child-like and enjoy pranks.
When you get onto Japanese stuff, though, they have entirely crazy monsters, like Tenjoname, the ceiling-licker. Apparently a mythological rationalization for water damage to ceilings, the tenjoname literally is a creature that wants to break into your house in the middle of the night to lick your ceiling (and do nothing else), leaving a slobber stain on it.
I'd honestly like to see some more of the utterly bizarre-but-benign magical creatures out there.