It has been done before. Saw pictures on the web years ago. Some Asian builders modded standard 3x3x3 cubes with chains. They hammered nails into the corners or edges of the cube, attached chains, and adjusted the length of the chain just so that the two corners could move freely around the cube but were restricted to always being adjacent to each other. On one cube, there were two sets of such corners. Endless variety of possibilities with chains, corners to corners, edges to edges, multiple chains on a cube might get kinked together etc. The cubes had a very demonic look to them.
Here's an idea to get the ball rolling: You could try it with bungees and have many pairs of pieces mated together. When you scramble it, would probably create a complex twisted jungle of entanglements. Think Brainstring + 3x3x3. Traditional solving algorithms will not work. Longer bungees and more pieces bandaged would allow deeper knotting before the cube binds up. I've not heard of anyone here trying it, but solving it would undoubtedly be a daunting task
My Creepy 3D Rubik's Cube Video
Yeah, Uwe is Dalai Lama and Paganotis is mother Teresa of Calcutta.