In The Once and Future King, what is Wart's quest?
I think if we had to specify a "quest" that Wart embarks upon, it is the quest of maintaining peace and stability in his kingdom through doing what is right rather than favouring the use of might to support his power. Consider the tutelage that Merlyn gives Wart where again and again, Merlyn tries to impress on his young student the abhorrent nature of war and the illusory concept of boundaries. Remember how Wart learns this lesson through watching geese flying in the sky. Merlyn points out that the geese fly wherever they will and do not heed boundaries that man places such importance on. As a result, land, and the possession of a tract of ground, is not worth shedding blood over. This is a lesson that King Arthur as an adult, on the wake of his final battle, evidently remembers, as he states his belief that war is something that is "fought about nothing--literally nothing."
Wart's quest then is to rule justly and honourably without falling into the tendency of using might to maintain his power. He, throughout his life, seeks to support the power of right against the tyrany of might.