What do you mean by dimentions?
In square 64 there would be 2 to the power 62 or approximately 5x10 power 17, or 5 with 17 zeros after it.
If you mean total rice grains then approximately 4.9 x 10 power 17 or 25 with 16 zeros after it.
The thing to remember with this is that total value is 1 less than the previous number. So 1, 2, 4, 8.16 added equals 31 or (16 x2)-1
If you want an exact number then get a big bit of paper you will need it!
If you fill a whole chessboard with rice, by putting one grain on the first square, 2 on the second, 4 on the third etc, doubling each time - then what are the minimum possible dimensions of the board?