Tom Bombadil is a nature spirit, or at least was before LotR was written (“Do you think Tom Bombadil, the spirit of the (vanishing) Oxford and Berkshire countryside, could be made into the hero of a story?”). The idea of ‘nature spirits’ seems a bit foreign to LotR, but it actually isn’t - while a lot of their traditional roles are filled by Maiar of various kinds (not to mention the Valie Yavanna), there is one key example of a genuine sentient piece of landscape: Caradhras. Ignore the films’ ideas about it being Saruman who attacked the Fellowship there - the books make it very clear that the Redhorn itself is out to get them. Admittedly the mountain doesn’t appear to be embodied, but the idea of the landscape having a personality is there.
So Tom is a nature spirit (probably). He has difficulty describing himself (“I am”, and his wife similarly says, “He is”), but when pressed calls himself ‘the Master’. Not the ruler of the land he personifies - the things in it, he says firmly, are not his - but its master. He can control everything in his lands, to the point where he can quell the ancient Huorn (or similar creature) Old Man Willow, and even drive out a Barrow Wight with no visible effort. He can seemingly teleport - he shows up instantly when Frodo calls from within the barrow, despite presumably being at home at the time. He’s also unaffected by the Ring - in the same way that a tree, presumably, wouldn’t turn invisible if you stuck it on a branch.
But the actual extent of Tom’s lands is a bit of a puzzler. He claims mastery over the Old Forest, including the Withywindle valley, and the Barrow Downs. But… why those limits? The boundary there is purely artificial - the Old Forest used to cover the whole of western Middle-earth, and even the Barrow-downs extend pretty much directly into the South Downs. Tom seems limited by the roads, rather than anything natural.
Which may be the explanation. As a nature spirit, the arrival of Men and all their works would be like a jab in the ribs. Every time Men put down another road, they were further binding nature to their will - and further binding Tom Bombadil to the wilderness that remained.
Okay, that actually is a new theory - but that’s not the one I was going for here. This is:
I think Tom Bombadil has a Palantir.
The evidence lies in the dreams the hobbits had while in his house. On the first night, Frodo dreams about Gandalf in Isengard, Pippin about trees surrounding the house, and Merry about rising water. Sam doesn’t dream at all. On the second night, we only hear about Frodo’s dream - of being on the Tower Hills, thinking about the Sea.
Obviously, we’re supposed to assume Pip is having nightmares about being trapped by Old Man Willow. But then… why is the house surrounded? And why is Merry - who was also trapped by the tree - dreaming about water?
My theory is that the Palantir - elven magic from Valinor - interacts with the presence of a powerful nature spirit to give people visions without them actually looking into it. It also appears to come unstuck in time, showing both the past and the future. We know, in fact, that the Palantiri are capable of seeing at least into the past – after Denethor dies, almost anyone attempting to use his sees only his burning hands, despite them being long gone.
Assume that Saruman was actually using his Palantir at the time the hobbits lay sleeping. If he was looking westwards - perhaps searching after Gandalf, who by then had escaped - he could have opened a link with Tom’s Stone. Frodo, too, was worrying about Gandalf - so the linked Seeing Stones showed him Gandalf’s most recent connection with Orthanc, specifically his escape. The other hobbits weren’t so focussed, so the Stones, influenced by Tom, showed them something else - their own future interactions with Isengard. In Merry and Pippin’s case, that is the attack of the Ents (trees surrounding the building, trying to break in) and the subsequent flooding (the rising water). And Sam? Sam never goes anywhere near Orthanc. He only passes it on the journey home. With no connection to draw on, he simply doesn’t dream.
Normally, of course, the Palantiri need to be held to be used. Bearing in mind that Bombadil is capable of, at minimum, mild hypnosis (the hobbits spend literally an entire day listening to him without even getting hungry, or noticing how long it's been), transmitting visions from the Palantir seems entirely in-character. It even seems he or Goldberry did so deliberately - all three hobbits hear (or think they hear) Bombadil's/Goldberry's voice at the end of their visions.
The next night, the Orthanc-stone isn’t open. Nor is the Anor stone, nor, fortunately, that of Minas Ithil - but the Palantir atop the White Towers, looking out over the Sea, is always active. Frodo - the only dream we’re told about - thus dreams about the Tower Hills, and the passage to Valinor which that Palantir shows.
But where did he get it? There were seven Stones originally, of course, of which four are still active - Orthanc, Anor, Ithil, and the Tower Hills. Of the remaining three, it would be highly unlikely for Tom to have gotten his hands on the Osgiliath Stone - it’s a very long way away, after all. That leaves Annuminas (capital of Arnor, later held by Arthedain) and Weathertop (on the border between the three later kingdoms).
We know that the princes of Cardolan - one of the three successor kingdoms to Arnor - were laid to rest on the Barrow-downs, which lay in their territory. We also know that the two lesser kingdoms - Cardolan and Rhudaur - squabbled over the Weathertop Palantir - since Arthedain already had one, they were rather less interested.
In 1409 of the Third Age, Weathertop fell, with its Palantir being retrieved. Shortly thereafter, the last Prince of Cardolan died, and the refugees retreated to the Barrow-downs and Old Forest. It is assumed that Arthedain took the Palantir, only for it to be later lost when Arvedui’s shop went down, hundreds of years later. But… what if it wasn’t? What if King Araphor of Arthedain - whose father had died in the failed defence of Weathertop, and thus secured the Palantir’s recovery - gave the Seeing Stone to the survivors of Cardolan in an effort to cement their alliance? After all, with Angmar-controlled Rhudaur to the east, he certainly didn’t want his southern neighbours to turn on his fragile nation as well.
So Cardolan takes the Weathertop stone - and when at last they retreat to the Old Forest, they take it with them. Two hundred years later, even the power of Tom Bombadil isn’t enough to protect them from the plague, and they die out - but the Palantir remains, and is retrieved by Tom.
And the fact that it is the Weathertop stone Tom holds makes the whole theory possible. The Tower Hills Palantir is ‘locked’ – it looks out over the Sea, and cannot be used to communicate with the other Stones. But the Stone at Weathertop was a ‘master-stone’, with greater powers than any of the others except Osgiliath. As the northern ‘master’ (the same title as Tom gives himself!), Tom’s Palantir, boosted by the power of the landscape itself, may well have been able to tap into the vision from the Tower Hills stone. Not to change it, of course – but to transmit it to a distant viewer.
A final thought. If Tom Bombadil does indeed have a Palantir, then he has the power to look out over all the lands of Middle-earth – all the country that was once part of his Old Forest, untamed wilderness which he walked at will. Now he is constrained to a tiny land, unable to reach his old territory – but still, through the Seeing Stone, able to watch it slowly decay under the influence of Men…