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We consider a simple symmetric random walk on a spider, that is a collectionof half lines (we call them legs) joined at the origin. Our main question isthe following: if the walker makes $n$ steps how high can he go up on all legs.This problem is discussed in two different situations; when the number of legsare increasing, as $n$ goes to infinity and when it is fixed.

How Tall Can Be the Excursions of a Random Walk on a Spider