EASIER TO READ ANSWER: I used a different format so it's easier to see the answers:

How many squares are in this puzzle?

Count all of the "obvious" squares first:
There are 16 squares arranged in a 4 X 4 pattern.
There are 8 little squares in 2 sets of 4.
You've found 24 squares so far: 16 + 8

Include the 1 large square surrounding the entire puzzle and the 2 squares surrounding each set of the little squares. Now you've found 27 squares so far: 24 + 3


Then look at the 16 squares again--each set of 4 (2X2) gives you another square, and there are 9 SETS of 4 squares, so now you have found 36 squares: 27 + 9

1 1 2 2
1 1 2 2
3 3 4 4
3 3 4 4

O 5 5 O
O 5 5 O
O 6 6 O
O 6 6 O

O O O O
7 7 8 8
7 7 8 8
O O O O

O O O O
O 9 9 O
O 9 9 O
O O O O

Each number from 1 to 9 shows the placement of the 9 sets of 4 squares, above.

Then, still looking at the 16 squares, each set of 9 squares (3X3) gives you another square, and there are 4 SETS of 9 squares.

Now you've found all of the squares: 36 + 4= 40 squares total

O 1 1 1
O 1 1 1
O 1 1 1
O O O O

2 2 2 O
2 2 2 O
2 2 2 O
O O O O

O O O O
O 3 3 3
O 3 3 3
O 3 3 3

O O O O
O 4 4 4
O 4 4 4
O 4 4 4

Each number from 1 to 4 shows the placement of the 4 sets of 9 squares, above