There is an infinite pool of balls and an infinite number of boxes. Let's randomly drop n balls into the boxes and study the number of occupied boxes. Classical limit theorems, the law of large numbers and the central limit theorem, are non-trivial, but known. Can we now prove the law of the iterated logarithm? I will show how to do this using tools that have much wider applicability. The talk is based on joint work with Dariusz Buraczewski (Wroclaw, Poland) and Alexander Iksanov (Kyiv, Ukraine).