In a joint work with Izumi Okada, we study the capacity of the range of a  simple random walk in three and higher dimensions. It is known that the order of the capacity of the random walk range in n dimensions has a similar asymptotic to that of the volume of the random walk range in n-2 dimensions. Proving the law of the iterated logarithm for the capacity of the range, we find that this correspondence breaks down for n=3, leading to an unexpected host of challenging open problems.