Bombadil

He is a strange creature, but maybe I should have summoned him to our Council.'

'He would not have come,' said Gandalf.

'Could we not still send messages to him and obtain his help?' asked Erestor. 'It seems that he has a power even over the Ring.'

'No, I should not put it so,' said Gandalf. 'Say rather that the Ring has no power over him. He is his own master. But he cannot alter the Ring itself, nor break its power over others. And now he is withdrawn into a little land, within bounds that he has set, though none can see them, waiting perhaps for a change of days, and he will not step beyond them.'

'But within those bounds nothing seems to dismay him,' said Erestor. 'Would he not take the Ring and keep it there, for ever harmless?'

'No,' said Gandalf, 'not willingly. He might do so, if all the free folk of the world begged him, but he would not understand the need. And if he were given the Ring, he would soon forget it, or most likely throw it away. Such things have no hold on his mind. He would be a most unsafe guardian; and that alone is answer enough.'

- The Lord of the Rings, The Fellowship of the Ring, The Council of Elrond

Tolkien scholar Michael Martinez has a very plausible explanation, and it has to do with the Ring's modus operandi - appealing to the personality of the individual:

"Given that Bombadil asked to see the Ring, and that he played with it and at one point had a gleam in his eye, I don’t see any justification for concluding that he was not tested by the Ring like others. Bombadil probably had the easiest test of all because he had already long before made his choice about mastery over others.

Bombadil allowed evil things to remain in his land — not because he wanted them there but because he did not want to destroy them. He probably set the boundaries of that land to keep those evil things from troubling Men and Hobbits (his neighbors).

Bombadil didn’t believe in creating prisons for the barrow-wights and... Old [Man] Willow; he just didn’t succumb to their evil ways. Hence, the Ring could have shown him a world where he roamed free and evil things didn’t bother him (and perhaps didn’t bother anyone else). Or the Ring could have shown him a world where he could “master” anyone and anything. The point is that the Ring definitely could have shown him something, even if it was more absurd and silly than what it showed to Sam.

And obviously, whatever the Ring showed Tom, he wasn't interested."

Tolkien explains why in Letter #144:

"Tom Bombadil is not an important person – to the narrative. I suppose he has some importance as a 'comment'. I mean, I do not really write like that: he is just an invention, and he represents something that I feel important, though I would not be prepared to analyze the feeling precisely. I would not, however, have left him in, if he did not have some kind of function.

The story is cast in terms of a good side, and a bad side, beauty against ruthless ugliness, tyranny against kingship, moderated freedom with consent against compulsion that has long lost any object save mere power, and so on; but both sides in some degree, conservative or destructive, want a measure of control.

But if you have, as it were taken 'a vow of poverty', renounced control, and take your delight in things for themselves without reference to yourself, watching, observing, and to some extent knowing, then the question of the rights and wrongs of power and control might become utterly meaningless to you, and the means of power quite valueless. It is a natural pacifist view, which always arises in the mind when there is a war."

So the thing that makes the Ring so appealing to almost everyone else is the same thing that makes it powerless against Tom Bombadil. He has deliberately chosen to reject control over anything but himself. He himself is good, but he doesn't take sides in the battle between good and evil. He doesn't care about who wins or loses, because he has renounced power, control, and conflict. These concepts have no meaning to Tom. He is exclusively an observer, and his only interest is in things for their own sake. Tom doesn't approve of Old Man Willow's attempt to eat the hobbits, and he stops him from doing so, but he still respects Old Man Willow and his right to exist. The fact that Old Man Willow is probably evil, at least to some extent, doesn't enter into the equation.

The Ring is a device of power, and power is its only means of influencing and manipulating people. Tom Bombadil is diametrically opposed to power, and therefore, the Ring is powerless to corrupt or tempt him. It is interesting (and amusing) to imagine the Ring desperately trying to find a chink in Tom's armor, searching in vain for a way to reach him, and silently screaming with rage when he treats the Ring like a silly little toy. Because the Ring tailors its appeal to the individual's unique weaknesses (i.e., desires), and because Tom has no desires (aside from keeping his wife Goldberry happy), the Ring simply couldn't reach Tom. He is entirely beyond its influence, and it is just another piece of jewelry to him.