Is it mentioned in the Sunnah not to repeat the request for the same thing more than three times, or to make an analogy to that, such as not knocking on a door more than three times?

The Prophet (peace be upon him) encouraged self-restraint from asking people, and it was reported that he made some of his companions pledge not to ask people for anything, to the extent that one of them would drop his whip and not ask anyone to hand it to him. Asking people is blameworthy, and insistence in it is even more blameworthy, due to the humiliation it causes the asker and the embarrassment it causes the asked. Whenever possible, refraining from asking people is better. If one is in need, they should ask gently, without causing harm or embarrassment. There is no mention in Islamic law of limiting asking to three times, and the prohibition of asking people is general for those who are self-sufficient. As for asking for money out of excessive desire, it is very blameworthy.