See, that's the fun part. If you try to pack in more information than given by these limits, you inevitably form a black hole (and then you lose access to those individual cm^3 regions, and your plan falls apart).
How so? Those 1cc-cubes all have the same exact density, and placing them together does not affect/change the density of the formed object (the 1m-cube).
For what you're saying, they would have to be so massive and dense (in the first place) that they would gravitate-in any matter placed near them (to then form a black hole).
But does that even hold true when you run the numbers to see if that "max-info" 1cc-cube is anywhere near the required mass (given it's volume of 1cc^3) to form a black hole?
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.
