The simplest way is to start by dissecting it by 4x4x4 = 64. If you then dissected one of these 64 cubes into 2x2x2=8 cubes you have increased the number of cubes by 7. You can repeatedly do this, either to more of the original 64 cubes, or even some of the new smaller cubes. Either way, dissecting 9 cubes into 8 will increased the number of cubes from 64 to 127.
In fact you can dissect a cube into almost any number of smaller cubes. There is just a short list of small numbers that cannot be achieved, the highest being 47.