Is it true that Allah does not accept a servant's repentance on the fourth time, and that this contradicts the encouragement for repeated repentance until Satan becomes weary?

There is no authentic hadith stating that Allah does not accept a servant's repentance after the fourth time; rather, authentic hadiths prove the opposite.

The hadith narrated about the wine-drinker, that Allah would not accept his repentance after the fourth time, is meant for extreme admonition and warning, and not to be taken literally.

Most narrations about the wine-drinker do not include "if he repents, Allah will not accept his repentance."

The hadith: "If a servant says 'I seek forgiveness from Allah and repent to Him'... Allah records him among the liars on the fourth time" is very weak (fabricated).

What the questioner mentioned about repeating a sin is a statement attributed to Ali ibn Abi Talib, may Allah be pleased with him, and not a hadith directly attributed to the Prophet, peace be upon him. Its meaning is that the servant should not tire of repenting and seeking forgiveness until Shaytan is the one who is vanquished.