Cubing the cube is the analogue in three dimensions of squaring the square: that is, given a cube C, the problem of dividing it into finitely many smaller cubes, no two congruent.
Unlike the case of squaring the square, a hard but solvable problem, cubing the cube is impossible. This can be shown by a relatively simple argument. Consider a hypothetical cubed cube. The bottom face of this cube is a squared square; lift off the rest of the cube, so you have a square region of the plane covered with a collection of cubes
Consider the smallest cube in this collection, with side c (call it S). Since the smallest square of a squared square cannot be on its edge, its neighbours will all tower over it, meaning that there isn't space to put a cube of side larger than c on top of it. Since the construction is a cubed cube, you're not allowed to use a cube of side equal to c; so only smaller cubes may stand upon S. This means that the top face of S must be a squared square, and the argument continues by infinite descent. Thus it is not possible to dissect a cube into finitely many smaller cubes of different sizes.
Similarly, it is impossible to hypercube a hypercube, because each cell of the hypercube would need to be a cubed cube, and so on into the higher dimensions.
Read more about this topic: Squaring The Square