For East Asian palaces, I pick Beijing Imperial Palace, for its perfect architecture, design, history and intact. I did not choose Changdeokgung because most of its complex already destroyed by Japanese invasion only garden is still intact. Its asymmatric layout is interesting but similar to Summer Palace.
For South Asia: Agra Fort. Actualy I think Agra and Red Fort is quite similar as fortress palace, in my opinion palace in Fatehpur Sikri is more interesting for multi culture design, but nothing can compete the art of Agra Fort, so its the best in the group.
European Palace, I choose Versailles and Schonbrunn as the grand daddy of all European Palaces, also for their history and politic roles. The Russian palaces in Kremlin and Saint Petersburg are among the best but not in the list.
Islamic Palace: I don't know why Alhambra is not in the list, maybe the inclusion of Albayzin Quarter made its not a pure palace anymore.