In one of Scott Aaronson's lectures he mentions that attempting to scale up any three or even two dimensionally laid out hard drive will eventually result in a black hole. That is, if you've got some thing you want more of in the smallest amount of space, if you keep scaling it up in the same layout, regardless of its density, if the density remains constant and you scale uniformly in either 3 or 2 dimensions, eventually it will form a black hole.
But he also points out that scaling it up in one dimension will not result in a black hole.
So, my question is: what is the greatest fractal dimension that a region of fixed density could indefinitely be scaled up that would never result in a black hole?
I suspect that dimension 2 might already be the critical value and that anything even slightly less than 2 would be safe from collapse, but I have no idea how to go about calculating or proving this.