  Counting the types of squares rather than their occurrences, we consider the
problem of bounding the number of distinct squares in a string. Fraenkel and
Simpson showed in 1998 that a string of length n contains at most 2n distinct
squares. Ilie presented in 2007 an asymptotic upper bound of 2n - Theta(log n).
We show that a string of length n contains at most 5n/3 distinct squares. This
new upper bound is obtained by investigating the combinatorial structure of
double squares and showing that a string of length n contains at most 2n/3
double squares. In addition, the established structural properties provide a
novel proof of Fraenkel and Simpson's result.
