Cover time of a random walk on a complete graph

A complete undirected graph has $N=100$ vertices. On the first turn you jump to a vertex chosen uniformly at random; on each later turn you jump to a uniformly random vertex, where staying put is allowed. What is the expected number of turns until every vertex has been visited at least once? Round to the nearest integer.

$519$. Each turn lands on a uniform vertex, so this is the coupon collector problem: with $k$ vertices seen, a new one appears with probability $(N-k)/N$, taking $N/(N-k)$ turns on average. Summing gives $N H_N=100\cdot H_{100}\approx518.7$.