What is Scout's nickname?In the book I can't find this answer.

The reason you cannot find the answer is that Scout's nickname is "Scout."  That is not the name she was born with, so this is a bit of a trick question!  Throughout most of the book, Scout is addressed only by this nickname.  However, her real name is given in the first chapter, provided by Miss Caroline, her teacher, on the first day of school. First, Scout explains to Miss Caroline,

Jem says my name's really Jean Louise Bullfinch instead of Finch (17). 

So we might think that Scout is joking and Jem has made up this entire name.  But as we read along, we see that Miss Caroline does address Scout as Jean Louise when she is trying to explain to Miss Caroline about Walter Cunningham's family background.  Miss Caroline, who is already clearly exhausted by Scout, asks,

What is it, Jean Louise? (20)

This is how we know this is actually Scout's name, the name in which she was enrolled in school.  I believe there are a few other references to "Jean Louise" in the book.  Perhaps you can find them.