Will -

First square 1 grain is 2⁰ grains
Second square 2 grains is 2¹ grains
Third square 4 grains is 2² grains
...
64th square is 2⁶³ grains

The sum of all of the grains on the board is 2⁶⁴ - 1 grains.

(I left the calculation of the sum as a homework problem for the students.)

John