How much data could I store on a Rubik's Cube? Google tells me that a standard 3x3x3 Rubik's Cube has 43,252,003,274,489,856,000 permutations. If I wanted to store data on that Rubik's Cube, how much could I store?

The only way I see to store data on a Rubik's Cube is to assign an integer to each permutation, then convert that integer into binary. It would take 66 bits to store an integer of that size, so the Cube could theoretically hold 65 bits of information by my calculations.

Is there any way to eke out more than 65 bits of storage from a Rubik's Cube?
If you consider the cube as living in a physical space, then you can get an extra factor of 24 from its orientation: assume it's sitting on a table and note the uppermost face (six options) and the face that's facing north (four more options). That gives you $\lfloor \log_2 (24\times 43\ 252\,...)\rfloor = 69\,\text{bits}$, and you could get even more by considering more fine-grained notions of orientation, at the risk of losing information by somebody jostling the cube.