In The Fellowship of the Ring, in the council of Elrond, Bilbo at first believes that everyone expects him to take the Ring to be destroyed, saying, "It is plain enough what you are pointing at. Bilbo the silly hobbit started this affair, and Bilbo had better finish it, or himself."
However, nobody expected him to go, and Gandalf told him, "...The word was meant, and we do not doubt that you are making a valiant offer. But one beyond your strength, Bilbo. You cannot take this thing back. It has passed on. If you need my advice any longer, I should say that your part is ended, unless as a recorder. Finish your book, and leave the ending unaltered! But get ready to write a sequel." Thus, Gandalf told him very gently that he was to old, and that he was no longer needed, except to write of the fellowships adventures.
Also, as Bilbo said, "I don't suppose I have the strength or luck left to deal with the Ring. It has grown, and I have not."
In a nutshell, Bilbo was to old to go on any more adventures.