On some aspects of the coupon collector's problem

Suppose that a company distributes a commercial product and that each package contains a coupon. There are $n$ types of coupons, and a customer wants to collect at least one of each. How many packages need to be bought on the average until getting all coupons? This is referred to as the coupon collector problem, and goes back at least as far as de Moivre.

Clearly, one expects that, in the beginning of the process, most coupons obtained will be new ones. As we continue, it takes more and more time to obtain a new coupon. We will start from the question what is the maximum time between two consecutive new coupons throughout the whole process. Then we will discuss some related problems.