One of the confusing issues here is that there is not a single, constant density necessary for the formation of a black hole. Instead, the Schwarzschild radius

One of the confusing issues here is that there is not a single, constant density necessary for the formation of a black hole. Instead, the Schwarzschild radius of a (potential) black hole is proportional to total mass. That means that as you increase its mass, the actual radius of a sphere of constant density grows only as m^(1/3) while its Schwarzschild grows much faster as m^1. Thus, as you pile up more and more of your identical 1cc cubes, the size of the pile will grow more slowly than the size of its Schwarzschild radius. As soon as you add enough cubes for the Schwarzschild radius to exceed the actual radius, the system must inevitably form a black hole. I think that this behavior is directly related to the information density limits that we've been talking about, but certainly the end result is the same: piling up lots of similar stuff in one place will eventually lead to gravitational collapse.