We should have enough LEDs around to make something like this…
“Looking Glass Founders See Their L3D Cube As A New Type Of Display”
I’ve seen things like this before as novelties that had some pre-programmed sequences they ran. I believe they were monochrome and probably didn’t have as many LED’s.
Looks to be 8x8x8 - a manageable number of components. Looks like they’re using addressable strips so you wouldn’t need 3D wiring.
Here’s a link to Instructables. Has some videos