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The Game of Chess

When the creator of the game of chess showed his invention to the ruler of the country, the ruler was so pleased that he gave the inventor the right to name his prize for the invention.
The man, who was very wise, asked the king this: that for the first square of the chess board, he would receive one grain of wheat (in some tellings, rice), two for the second one, four on the third one, and so forth, doubling the amount each time. The ruler, who was not strong in mathematics, quickly accepted the inventor's offer, even getting offended by his perceived notion that the inventor was asking for such a low price, and ordered the treasurer to count and hand over the wheat to the inventor.
However, when the treasurer took more than a week to calculate the amount of wheat, the ruler asked him for a reason for his tardiness. The treasurer then gave him the result of the calculation, and explained that it would be impossible to give the inventor the reward. The ruler then, to get back at the inventor who tried to outsmart him, told the inventor that in order for him to receive his reward, he was to count every single grain that was given to him, in order to make sure that the ruler was not stealing from him.

If a chessboard were to have wheat placed upon each square such that one grain were placed on the first square, two on the second, four on the third and so on, doubling the number of grains on each subsequent square, how many grains of wheat would be on the chessboard at the finish?
Note that a chess board is an 8×8 square, containing 64 squares. If the amount doubles on successive squares, then the sum of grains on all 64 squares is 18,446,744,073,709,551,615 grains.
In terms of volume: if it is assumed that a grain of rice occupies a volume of 2 cubic millimetres then the total volume of all the rice on the chess board would be about 36.89 cubic kilometres.