If I charlieplex an LED cube, using 0603 LEDs, it could look like this:
Pictures pulled out from my archive, where I've been toying around with this idea since the last Square Inch Contest, three years ago.
That's pretty dense, but the majority of it can be made layer-by-layer, like this:
That means each layer looks like this:
A couple years ago, I laser cut a stencil and soldered up one layer, and it seemed to work okay. Not easy, but it was possible.
I never did multiple layers, and the hard part would be getting those vertical wires attached, near the centre of the cube. I have an idea, though.