The minimum number of squares a 13 by 13 square can be dissected into is 11:

There is one trivial solution to the size of the smallest square which can be dissected into squares which are all different sizes: the unit square! The smallest non-unit square which can be dissected into squares which are all different sizes has sides of length 175 units.

For further information on this and other similar problems see Chapter 11, Mrs. Perkins Quilt and Other Square-Packing Problems in Mathematical Carnival by Martin Gardner, published by Pelican books.