What did Bilbo Baggins do while the Fellowship of the Ring went on their quest?

Question

Bilbo is responsible for leaving the One Ring for Frodo, who subsequently became hunted by the Nazgûl, Saruman, Sauron, and even Boromir in a way. Yet in the Peter Jackson film Fellowship of the Ring, we come across Bilbo in Rivendell kind of just hanging out -- he gives Frodo all his weaponry and armour, tells Frodo he is sorry he brought Frodo into the whole One Ring mess, and then basically bids Frodo adieu and watches as Frodo sets off on the Fellowship of the Ring quest.

Almost all races are involved in the war to destroy the Ring. But what does Bilbo do during the quest of the Fellowship? Does he simply remain in Rivendell and work on his memoir? Is there any information regarding Bilbo and what he did while the quest of the Fellowship of the Ring was ongoing? What did he do during the time the One Ring was being taken to Mt. Doom?

_{ETA: The usual caveat. I've read The Fellowship of the Ring and The Two Towers, but have not read The Silmarillion or The Return of the King. I've seen Peter Jackson's Lord of the Rings Trilogy many times. I just got The Hobbit: An Unexpected Journey on BD this past Tuesday, so I'm pretty unfamiliar with it still. I.e. I'm not a Tolkien scholar, so please pardon me if this is an obvious question to those in the know.}

Answer 1

Advance warning: I'm going to assume that the fact that you've asked this question, despite having not read RotK, means that you don't mind being spoilered on anything in RotK.

If this assumption is false, then stop reading this answer and start reading RotK. Now.

End of advance warning.

When the Hobbits get back to Rivendell, Bilbo has grown old. He's sleepy, he doesn't get much done, he feels the end of his life coming on.

They found him all alone in his little room. It was littered with papers and pens and pencils; but Bilbo was sitting in a chair before a small bright fire. He looked very old, but peaceful, and sleepy.

From Bilbo himself:

"Do you know, I shall be one hundred and twenty-nine? And in one year more, if I am spared, I shall equal the Old Took. I should like to beat him; but we shall see."

Some general behaviour:

...they sat much with their old friend, who spent most of his time now in his room...At first he pretended to take some notes; but he often fell asleep; and when he woke he would say: 'How splendid! How wonderful! But where were we?' Then they went on with the story from the point where he had begun to nod.

He gets forgetful:

Then he gave Frodo his mithril-coat and Sting, forgetting that he had already done so ... 'what's become of my ring, Frodo, that you took away?'

His poem:

The Road goes ever on and on
Out from the door where it began.
Now far ahead the Road has gone,
Let others follow it who can!
Let them a journey new begin,
But I at last with weary feet
Will turn towards the lighted inn,
My evening-rest and sleep to meet.

Sam's observation:

'I don't think, Mr. Frodo, that he's done much writing while we've been away. He won't ever write our story now.'

Bilbo's reply:

'You see, I am getting so sleepy,' he said. 'And when I have time to write, I only really like writing poetry.'

And much much more.

This is not a Hobbit who is going to do anything while the Fellowship were away. This is an old Hobbit at the end of his life. If all that you have to go by is Jackson's movie, you're not going to get the full weight of this. So I'll end as I started: read RotK.

Answer 2

Bilbo had given the ring up almost two decades before and though he reacted strongly to it when it reappeared, he recovered his composure. The books are infused with the idea that Hobbits are for some reason less affected by the One Ring's miasma--not unaffected, but resistant.

Further he was an old, old Hobbit and was aware of the demands of questing and his own limitations.

Bilbo had for a long time been contented with his maps and his writing, he was presumably happy as a clam living in Rivendell, and I see no reason to suppose that he did anything more than bug Elrond for updates rather more often than they were available.

Answer 3

I can't improve on dmkee's answer, but I'll add this just for completeness' sake:
(By the way, this is from Chapter 1, right after the Party)

[Bilbo:] "Well I've made up my mind, anyway. I want to see mountains again, Gandalf – mountains; and then find somewhere where I can rest. In peace and quiet, without a lot of relatives prying around, and string of confounded visitors hanging on the bell. I might find somewhere where I can finish my book. I have thought of a nice ending for it: and he lived happily ever after to the end of his days."

and this is one or two pages later:

"I am being swept off my feet at last."

This seems to suggest that he wants to go out walking again, and then go to Rivendell, and consume copious amounts of Lembas (see dmkee's answer).

I also found this in the Tale of Years (Appendix B):

3001 Bilbo's farewell feast [september]. Gandalf . . .
3002 Bilbo becomes a guest of Elrond, and settles in Rivendell.

That gives us a rough timeline.