Unfortunately, even assuming a board diameter of 2, storing that game's any state would require some 10^435 times more bits than there are cube Planck lengths in the observable universe.
Your far, far, FAR bigger problem is UI. :) Again, you need not display all dimensions at once, so you can treat it as an n-dimensional problem, but that still calls for some crazy UI pretty quickly.
Still, it wouldn't have to. Since most arrays would have pieces with lower numbers( 2,4,8 ) you could very efficiently compress the data.
At some point you would run out of memory but it would be playable. Just like those Game of Life implementations with seemingly infinite grid.
BTW you are right about compressing: you could store the initial state very efficiently, and probably even play the to the end without trouble.