I thought of the following probability problem, which seems to have an answer of 1/e, and wonder if someone has an idea as to how to prove this.

Suppose a man has a bottle of vitamin pills and wishes to take a half pill per day. He selects a pill from the bottle at random. If it is a whole pill he cuts it in half, takes a half pill, and puts the other half back in the bottle. If it is a half pill, he takes that. He continues this process until the bottle is empty. What is the expected maximum number of half pills in the bottle? If the bottle starts with n pills, and M is the expected maximum number of half pills, then M/n appears to tend to 1/e as n tends to infinity.