Is a man allowed to propose more than four times, or does it mean that he is permitted to keep a maximum of four wives?

It is not permissible for a man who is married to four women to propose to or marry a fifth. If he does so, the marriage contract is null and void. The proposal itself is forbidden because it is a مقدمة (precursor) to an unlawful marriage. Therefore, your uncle is not permitted to propose to a fifth woman or contract marriage with her unless he divorces one of his four wives and her 'iddah (definition) (waiting period) expires. If he were to do so, he must repent, and the contract is null and void, and her divorce is of no effect. Whoever has intercourse with her while knowing the marriage is invalid is considered an adulterer.