Since the box is cube-shaped with each edge length being 20 inches, the volume of the box is calculated as 20 x 20 x 20 = 8,000 cubic inches.
To find the greatest possible volume of all the presents and packing material inside the cube, we need to fill the entire volume of the box.
Assuming that the packing material takes up some space, let's say half of the total volume or 4,000 cubic inches, the remaining 4,000 cubic inches would be the maximum possible volume for the gift items.
Therefore, the greatest possible volume of all the presents and packing material inside the cube is 4,000 cubic inches.