Richest as in best, one of a kind. Note the structure here:
 Some of his friends are rich and some of his friends are very rich. His sister was staying with one of his richer friends, with Max, at Max's house.
 All of his friends are very rich, but one is richer than all the rest. His sister was staying with his richest friend, with Max, at Max's house.
Hope that helps.
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