The unknown wheat-and-chessboard problem

You can read the original story of the Wheat and Chessboard problem on Wikipedia. (But you won’t get anything useful from it :))

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An eternity after the original creation of chess, a score of Swedish men re-invented the game, which was laid out in the warped space of the 378547252^nd dimension, and has been known as the game Chess 378547252. The board of Chess 378547252 was so large, that no one was able to count how many cells (in the grid) there were.

The same old king, however, was delighted to find it enjoyable, and decided to award the creator(s) of Chess 378547252 whatever they wanted.

– O, lord. Please, allow us to humbly request the smallest prize, of one grain in the first cell, two grains in the second cell, four grains in the third cell, and twice as many grains in each cell as that in the previous cell thereafter.

– You greedy men! I will have given you 18,446,744,073,709,551,615 grains, two thousand times as much as the annual total production in the world, after only the first 64 cells! And look, there are so many cells ahead. Do you think I’m a fool?

– No, lord. No. Please calculate this carefully: If we are given ONE EXTRA grain, we will have been awarded nothing at all!

The king looked at the formulae, verified it a thousand times so as to be absolutely sure it’s correct, and finally admitted that the men were right. In addition, the king decided to give the extra grain than the men originally asked for:

– Take this last grain, and you have zero with you. Now get out of here fast!

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No one ever since has known how the story continued, and scientists have worked hard its legitimacy and truthness.

You, the smartest user from Puzzling SE, are here for the challenge.

• How many grains were the men awarded, before the last one?
• Can you find out how is the story sensible?