Having a hard time dealing with the archetypal significance for this story.
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The archetypal significance of Bilbo's finding Gollum's ring actually more than one specific thing. Bilbo's descent into the mountain is th first archetype, and it symbolizes a descent into hell, ot the discovery of the darkest part in our own selves. Bilbo's reward for conquering the "darkest" place and discovering his true self is the discovery of the ring. He has no idea of its significance until many years later when he sees the changes in himself that have been brought on him by possessing the ring.
The second archetype is his own dark shadow, which is represented by Gollum. It is because of the ring, and due to his treatment of Gollum that Bilbo loses his innocence. He uses trickery, he lies,and he cheats; he even lies to Gandalf who he loves. When he sees Gollum, he sees what he could become if he allows to evil of the ring to overwhelm him.
Smaug's lair......... can be considered the archetypal quest for heroism. Bilbo defeats him, but he also returns the Arkenstone.By reaching his full potential, Bilbo no longer needs the Arkenstone as he's already gained a more valuable treasure; that being his true self.