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.