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Bilbo Baggins, Frodo's uncle and friend, is the one who starts this whole adventure with his discovery of the Ring in one of the tunnels under the Misty Mountains. Since The Hobbit starts the War of the Ring narrative, there is something nice about the fact that Bilbo reappears right at the end of The Return of the King. We keep mentioning that Tolkien has a great sense of continuity and plot structure. Everything that comes up at the beginning of the series winds up turning up again by the end. And Bilbo is no exception.
In fact, Bilbo comes up not only at the end of Book Six, but also at the end of Book Five. As Pippin loses consciousness at the Black Gate of Mordor, he sees the Eagles arriving at the battlefield. He thinks to himself, "Bilbo! […] But no! [The Eagles] came into his tale, long long ago. This is my tale, and it is ended now" (5.10.61). Luckily, Pippin is wrong that his tale "is ended now"; he recovers well at the Houses of Healing.
But Pippin's connection to Bilbo's history is totally spot on. The Eagles come at the end of both the Battle of the Five Armies in The Hobbit and at the fall of Sauron in The Return of the King, when they rescue Frodo and Sam. Again, we are amazed that Tolkien finds a way to carry this image from The Hobbit right to the end of The Return of the King. This dude is one masterful writer.
In Book Six, when Frodo meets with Bilbo again at Rivendell, Frodo finds that the health of his favorite uncle has started to decline. Bilbo becomes easily confused, and often can't remember exactly where (or when) he is. There is something deeply sad about Bilbo's questions about whatever happened to his old Ring. He does not understand that Frodo has just been on a quest to destroy it, nor does he remember the threat that the Ring represents. He just hasn't got a clue.
But Bilbo's forgetfulness and sleepiness also give him a break from the sadness he might feel if he learned what really went down. Bilbo has earned his retirement, since he is beyond old by both human and hobbit standards (129!), and everyone agrees that he deserves a rest. Bilbo may not be totally healthy as an old man, but even his illness proves that he has lived a long and healthy mortal life up until now, unlike the stretched, miserable experience of that other Ring-bearer, Gollum. While it is sad to see Bilbo passing, we are also pleased that the Ring has not had the kind of lasting bad effects on him that it could have had.
As a former Ring-bearer, Bilbo has earned the right to sail into the West with Elrond, Gandalf, and his own nephew. And so Frodo and Bilbo end their adventures by leaving Middle-earth, bringing an end to the events that began eighty years before, with a riddle game in the dark (see The Hobbit Chapter Five).