>>5
There are aperiodic square tiles, i.e. Wang tiles. The Wikipedia article on them has references on their use in texture generation:
http://en.wikipedia.org/wiki/Wang_tile
The reason they aren't useful is because it takes a lot of processing power to compute the tiling. If I tell you, get me the tile at (25,35). What is it? You've either got to do a whole lot of work to calculate it, or you have to store the whole grid on disk and load it into memory. If your grid is small of course the overhead is negligible, but then what was the point of using aperiodic tiles?