tolkiens legendarium - What happened to Tom Bombadil when the last ship left for the Undying Land?

When the wizards and elves leave Middle Earth the only great power that is left will be Bombadil. Towards the end of The Return of the King, when Gandalf leaves the hobbits, he mentions that he wants to have a long talk with Bombadil.

‘But if you would know, I am turning aside soon. I am going to have a long talk with Bombadil: such a talk as I have not had in all my time. He is a moss-gatherer, and I have been a stone doomed to rolling. But my rolling days are ending, and now we shall have much to say to one another.’

So what became of him?

Answer

Letter 144:

And even in a mythical Age there must be some enigmas, as there always are. Tom Bombadil is one (intentionally).

The end of Lord of the Rings is not a traditional wrapping-up of loose-ends and Bombadil is one that is (perhaps intentionally) left hanging. There's no mention of a subsequent history for him in either the Letters or the HoME volumes; that doesn't mean that such a subsequent history doesn't exist, of course, just that Bombadil's part in the story was done.

The nearest hint we get is in Gandalf's response to Frodo shortly after the passage you've quoted:

'As well as ever, you may be sure,' said Gandalf. 'Quite untroubled and I should guess, not much interested in anything that we have done or seen...'

Bombadil's part in the world is not to be involved in the Great Events; he doesn't get involved and they don't interest him. I'd even dispute that he's a "great power": he's very much a stay-at-home who doesn't get involved. This is also well-summarised by the discussion of him at the Council of Elrond:

'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 And now he is withdrawn into a little land, within bounds that he has set, though none can see them, waiting perhaps for a change of days, and he will not step beyond them.'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.'

So Bombadil is definitely someone who doesn't get involved, and any subsequent part he may have to play will most probably be along the same isolationist policy.