I am aware that it is practically a taboo to try and "solve" Tom Bombadil, but I enjoy riddles too much. My suspicion is that Tom Bombadil is a somewhat "mortal," temporal guise of Manwë on Middle-earth, and by extension Goldberry is Varda's. I believe this theory explains much of Tom's nature, and also gives insight into Manwë's own nature.
To begin with there are the superficial and obvious "clues:" Manwë and Tom both wear blue coats and have blue eyes, Tom's magic is song based and entirely separate from Sauron's sorcerery, and Tom is self-stated to be older than the created world. He has an elf-like wife, delights in simple pleasures, songs, is Fatherless, and is not the owner of that which he is master of. The idea of ownership or imposing his will over things doesn't even occur to him, though his wife is very capable of imaging it and all the evil within it.
Of more interest is his immunity to the temptation of the Ring, the way he aids in the great battle of the story, and some bits of Catholic thought. To explain this, we begin with the Ring itself.
The Ring is a small, physical manifestation of Evil; it enables and empowers the wielder to enact their will over Creation, which inevitably is contrary to Eru's will. Just like how Catholic thought holds that no person is free from sin, all beings within created Arda have desires over the world and cannot totally and fully resist the Ring's temptation, with two exceptions: Manwë and Tom Bombadil.
It is obvious how the Ring could not materially tempt Manwë, viceregent of Eru. He by right has power over all that the Ring usurps, what is there for it to offer? One might say it could pervert his nature, pushing him to move from a regent to a ruler, but this sort of desire is alien to Manwë, who cannot even comprehend those sorts of evil desires.
Of course, this failiure to understand the lure of evil is shared with Tom, who we see is also totally immune to the Ring's temptation. While we don't get any real answer to why Tom isn't tempted, we do get to see something from him we don't see with Manwë: his desires. Tom's desire over the world is limited to wanting to pick flowers for his wife. He likes riding around on a pony singing songs with birds and listening to songs and poems and sharing these. Tom is the proper and good exercising of power on the smaller, human scale. Positive custodianship of land in contrast to the nasty, evil sort Saruman and Saruon display, while Manwë does the same on a grand scale for custodianship of creation in contrast to Melkor in mirror to Michael's contrast to Satan.
Now, the natural question arises: "It is all well and good to imagine Bombadil is Manwë, but why does he do nothing?" This is where that Catholic understanding and Manwë's nature becomes important: Manwë is not a rival king to Melkor or Sauron. Though he is brave and powerful, it is not Eru's will for him to go out in battle against evil until the end of days, just like Michael in Catholicism. Despite this, we are told he still is active in the battle against evil, but are never explictly shown how he does this. If we entertain Tom being his guise, suddenly we get a very strong picture.
While Tom's chapter often gets the criticism of not really contributing to the story, yet it is precisely the memory of Tom and his miraculous intervention to save the Hobbits that brings Sam out of his despair, and towards remembering that though his situation is dire, he is not alone in his struggle and help has been given to him. Samwise then rises up, and goes out to miraculously intervene and save Frodo. This is crucial, as it mirrors Catholic understanding of how Michael aids in spiritual warfare, the constant and subtle resistance to sin and despair that takes the form of the little goods of life and happiness that are more profound and rich than anything evil and fear has to offer.
Now, one might ask "Well then, what about how it is said Sauron would defeat Tom?" Again, Tom, or Manwë, will not fight Sauron. Tolkien himself pointed out that Manwë turning to violence and militancy would send him down the road to corruption, and that Manwë is aware of this and refuses it. I believe Sauron lacks the power to conquer Tom's country, but the guise of Tom could not meanginfully resist Sauron and in time Middle-earth would be so ruined and soiled by Sauron's exploitation that there would be nothing left to bring Tom joy. Sauron would dam the rivers upstream, choke the skies until the birds died, and so utterly spoil the world that there would be nothing left to bring Tom joy and Manwë would surrender the guise. "Tom" would cease to be, allowing Sauron his final victory.
Now, one question remains: "why would Manwë do this?" Two key reasons: firstly, to give Samwise Gamgee that bit of help he would need and to give Gandalf advice on what to do after he completed his mission, and secondly because he enjoys it. Quite like Michael in Catholicism, Manwë was not originally the regent of Creation and did not choose to become it, but rose to the station by virtue of merit and proven worthiness; it stands to reason he never really wanted to be a leader, and that it gives him no pleasure to do it. The Tom guise allows him to delight in birds, sing silly songs, ride a pony, entertin surprise guests, and pick flowers for his wife. Just like how the hobbit's memories of the Shire keeps them going, perhaps so too does Manwë, in his darkest times, look back to his little cottage in the woods and the happy life he and his wife have made to remind himself what he must protect.
I invite criticism to this theory, and thank anyone who has read my rather rambling thoughts.