What is the most number of distinct free polyominoes you can form on the faces of a standard 3x3x3 Rubik's cube? Here a polyomino is considered as a set of orthogonally-adjacent cells of the same colour lying on the same face of the cube. Two free polyominoes are considered distinct if they are not a rigid transformation (translation, rotation, reflection, glide-reflection) of each other.
No pictures right now, sorry.
We have 9x6=54 squares to work with.
The maximum we can fit is:
1 monomino = 1
1 domino = 2 (total 3)
2 trominos = 6 (total 9)
4 tetrominos (line cannot fit) = 16 (total 25)
This leaves 29 squares left, which can fit at most 5 pentominoes or bigger.
So our theoretical maximum is 13.
We can do this with e.g.
U pentomino + T tetromino
L pentomino + square tetromino
L tetromino + extended square pentomino
W pentomino + L tromino + monomino
S tetromino + line tronimo
Domino + anything else that fits in the 7 remaining squares.
Here is an example of the six faces tiled with the maximum number of distinct free polyominoes, ie 13.
Method: I tiled a 3x3 with all possible combinations of polyominoes of size 1 through 7. Then I searched for all disjoint cliques and discarded several with only 12 pieces, and rendered just one of them.
Update: I re-ran the whole thing with octominoes included to get a count of all possible solutions. There are 78 ways of selecting 6 sets of pieces for the faces. 14 of them use 13 polyominoes, the rest 12. If you allow one of the faces to be a 3x3 there are 325 sets of polyominoes in total.
I'll leave the original answer here as an example of tiling the faces of a cube 'with wrap': Here is a 3-cube tiled with all polyominoes to size four plus five pentominoes. I tiled a net of a cube, you just have to fold it up in your imagination. 14 polyominoes total.
Method: I ran my tiler on this shape and told it to use up to five of the 12 pentominoes plus all of the smaller polyominoes. It's a zillion lines of hieroglyphic code designed for all sorts of other things so I won't post it here...