First, there is no rule against more than one verb in a clause.
I went to the barn and fed the cows.
Second, there is only one verb in your main clause -- "like".
"Staying is a gerund that functions as noun. "To watch" is an infinitive. "Rotate" is part of a reduced relative clause: "(that) rotate around the northern sky or a bare infinitive (no to). At least, that is my take on it.
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