'I am leaving NOW. GOOD-BYE! -- the shift in narrators from Bilbo to Frodo in 'A Long-expected Party'.

'Good night, Frodo! [said Bilbo] Bless me, but it has been good to see you again! There are no folk like hobbits after all for a real good talk. I am getting very old, and I began to wonder if I should live to see your chapters of our story. Good night!' (21 October 3018)

            FR 2.i.238

The evening deepened in the room, and the firelight burned brighter; and they looked at Bilbo as he slept and saw that his face was smiling. For some time they sat in silence; and then Sam looking round at the room and the shadows flickering on the walls, said softly: 

‘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.’ (5 October 3019)

            RK 6.vi.987

Both these scenes take place in Rivendell, not quite a year apart, with many a hard day in between for Frodo and Sam. In those long months we can see that a shift has occurred in whose story it was Bilbo was to tell. For Bilbo 'our story' is either Bilbo and Frodo's story, with separate chapters of course, or perhaps 'our' refers to Hobbits more broadly, as one of Bilbo's prospective titles for the story suggests, all of which were crossed out ('Adventures of Five Hobbits': RK 6.ix.1027). We might well wonder if the idea of calling what he suffered an adventure stuck in Frodo's throat. For Sam, in the second passage, 'our' refers mostly to him and to Frodo. Though he would never begrudge Captains Meriadoc and Peregrin their share of the credit, he cherished the honor he knew his master deserved and had no objection to the blush of glory himself especially when Rosie Cotton was within earshot (RK 6.viii.1014, 1016). 

Bilbo's list of tentative, struck-through, suggestions for the title of the work encompassing the stories of Bilbo, Frodo, Sam, Merry, and Pippin, stands in stark contrast to Frodo's grand and decisive title, seemingly conceived and executed in one go, which puts what he saw as most important boldly first and subordinates Bilbo and himself into, as it were, the metadata (RK 6.ix.1027). Whether it was Bilbo or Frodo who crossed out Bilbo's titles is uncertain, but that someone felt it right to do so (Frodo, I think) fits in very well with the Sam's impression that Bilbo hadn't gotten far at all in their absence. There are several other pieces of evidence I find quite telling in considering how much Bilbo might have written. 

1. Bilbo never changed his original version of the story of how he came by the Ring. Though he admitted it to Gandalf, to Frodo, and subsequently to all those present at the Council that he had lied about Gollum and the Ring, he didn't revise what we know as Riddles in the Dark to reflect the truth. The true version at first existed only '(as an alternative), derived no doubt from notes by Frodo and Samwise' according to the author of the Prologue, who believes Frodo and Sam could not bring themselves to alter what Bilbo had already written (FR Pr. 12). Bilbo's failure to incorporate the truth shows just how much a hold the Ring still had on him many years after he had given it to Frodo. This is especially telling given the very ugly moment he had with Frodo in Rivendell when he reached for the Ring, as a result of which he said that he understood about the Ring now (FR 2.i.232).
2. Bilbo was very keen to hear all the gossip from the Shire, and, however much he loves being in Rivendell, he misses being around hobbits. Given this and the great attention he gave to the preparations for his birthday party and his farewell presents, we can safely assume that he wanted to hear everything there was to hear about the reactions of his friends, relatives, and neighbors to his disappearance. The legend of 'Mad Baggins' must have given him quite a laugh.
3. The narrator of A Long-expected Party is very much like the narrator of The Hobbit, intrusive, humorous and prone to parenthetical asides, but his wit and persona vanish the moment Bilbo puts on the Ring and returns home to a fierce confrontation with Gandalf about leaving the Ring behind. The morning after Bilbo's departure the humor and asides return. I have discussed the use of parentheses in this chapter and in the rest of The Lord of the Rings in a series of posts beginning here but not yet completed.
4. In Letter 151 (p. 186) Tolkien says 'Frodo is not intended to be another Bilbo. Though his opening style is not wholly un-kin. But he is rather a study of a hobbit broken by a burden of fear and horror — broken down, and in the end made into something quite different.'

In view of this evidence, and of the evidence I have so far considered in my series of posts on the narrator's use of parentheses, I have come to the opinion that Bilbo indeed wrote very little of The Lord of the Rings. I think we might descry the limits of his involvement in the opening of A Long-expected Party up to his disappearance from that party -- which he would have found as great a delight to write as we find it to read -- but I believe he disappears as narrator the moment he vanishes from the party. If he could not bring himself to replace the lying account of the riddle game with the truth -- something even Frodo and Sam acquiesced in since it was already written -- how could he bring himself to face the ugliness of his confrontation with Gandalf over the Ring? From the moment Bilbo says '... this is the end. I am going. I am leaving NOW. GOOD-BYE! (FR 1.i.30), he is gone. Frodo picks up from there, with an 'opening style ... not wholly un-kin' but marked by 'the burden of fear and horror' he had lived through. And we can see this clearly in the juxtaposition of the party after Bilbo left, with the traumatic account of Bilbo's argument with Gandalf, and the reassertion of humor and Gandalf's warnings the next day.