Excuse me for not having read the books, I just rewatched the movies while beginning The Silmarillion, with The Hobbit and The Lord of the Rings to follow.
At the end of Return of the King as he leaves for Valinor Frodo gives the book containing The Hobbit and The Lord of the Rings to Sam, saying something along the lines of "the last pages are for [him] to fill."
Do we know what becomes of that book? Did it stay in the Shire, or follow Sam to Valinor? Was it reproduced to be held in the libraries of Elves and Men? Obviously, some of the stories would travel as news of the land, but what of the book itself?