9 pieces
An infinite number of ways. Cut along a line from anywhere on a side to the centre of the square. Make three more cuts, at 90, 180 and 270 degrees to the first at the centre. Each point on a side of the square will give rise to a different set of four identical pieces of the square. And there are an infinite number of points on the side of the square. So an infinite number of answers to the question.
A circle has an infinite number of lines of symmetry. As long as the line passes over the center point it cuts the circle in two identical halves.
Make two cuts normally, then make the third cut along a line that crosses both of the first two cuts but does not pass through their intersection.
An angel bisector is the line that cuts an angel exactly into two equal pieces! :]
Providing that the cuts dont overlap each other then the circle will be divided into 8 parts
21 cuts required to cut a cube into 504 identical pieces.
216
7
If you do not re-stack the pieces, then 15 cuts.
If no cut intersects any previous cuts, then you can just slice it into 14 pieces.
324
26
In one sense you cannot. The cakes would have a different number of faces which were part of the original faces. To that extent the pieces will not be identical. If such pieces are considered identical, and if the cake pieces can be stacked before cutting, then 9 cuts will suffice. Without stacking, 12 cuts are required. If the cake can be stacked and cut, and a little wastage (less than 2.5%) is pemitted, then 7 cuts will be enough.
26
7
An infinite number of ways. Cut along a line from anywhere on a side to the centre of the square. Make three more cuts, at 90, 180 and 270 degrees to the first at the centre. Each point on a side of the square will give rise to a different set of four identical pieces of the square. And there are an infinite number of points on the side of the square. So an infinite number of answers to the question.
Eight. Use the first two cuts to cut the pie into four pieces, then use the third cut to slice the top from the bottom, doubling the number of pieces to 8.