Radio Astronomy Systems Research Group

He can deliver 833 apples.

Break the trip into three legs. The first leg is 333 miles. Starting with 1000 apples, he will finish this leg with 666 apples. He returns to get another load and repeats this leg two more times. He now has 3*667 or 2001 apples. He starts on the second leg, which is 500 miles long. After completing this leg once, he goes back for the rest of his apples. Now he will have 2000 - 2*500 apples (sadly, he has to leave 1 apples at mile 333), and he only has 167 miles left to cover with a full load of apples. He will arrive with 1000-167 or 833 apples.