How has Bilbo Baggins stayed the same by the end of The Hobbit?
This is a really interesting question, because most discussions of the character of Bilbo refer to the extent to which he is a dynamic character and discuss the many ways that he has changed throughout the novel. Certainly, this differences between Bilbo at the beginning and Bilbo at the end are more apparent. However, if I were you, I would respond to this question by talking about the information we are given in Chapter One about the heritage of Bilbo and how he comes from a long line of hobbits who do strange things like go and have adventures. Note what we are told about his family background in this initial chapter:
It was often said (in other families) that long ago one of the Took ancestors must have taken a fairy wife. That was, of course, absurd, but certainly there was still something not entirely hobbitlike about them, and once in a while members of the Took-clan would go and have adventures. They discreetly disappeared, and teh family hushed it up; but the fact remained that the Tooks were not as respectable as the Bagginses, though they were undoubtedly richer.
Arguably, through the course of the novel, Bilbo Baggins uncovers his true Took-like nature, and discovers that he, like his ancestors, loves being unrespectable and having adventures. The way that at the end of the novel Bilbo quickly tires with the mundane and monotonous existence of his former life shows that he remains the same in this aspect. His Took nature dominates him and will not be ignored.