It was once believed that 6x6 and higher Rubik's Cubes would be impossible to physically make. In the real world, the outermost edges have to be a little thicker or the faces rounded in order for the puzzle to stay together while being turned. Nowadays they're widely available on Amazon.

The Megaminx puzzle is a dodecahedron, having 12 faces. Would it be possible to make a similar puzzle for an icosahedron? What about the stellated polyhedra? Or semi-regular polyhedra? There's no reason we have to stick to the platonic solids, right?

I'm no expert in 3d geometry or twisty puzzle manufacturing but maybe this helps?