A farmer has to make 8 stops in delivering eggs. He begins with exactly the number of eggs he needs for these 8 deliveries. At the first stop he delivers half of the eggs he has plus 1/2 of an egg. At each of the next 7 stops he delivers half of the remaining eggs plus 1/2 of an egg. When he is finished he has no eggs left, and none have been broken when making the deliveries. How many eggs did he have when he began his deliveries?