One thing that is crazy because of the banishment of the Princess of Sweet Rhyme and the Princess of Pure Reason is King Azaz’s banquet. The grand feast seems to unfold in a way that lacks rationality or common sense. When Milo suggests “a light meal,” the waiter doesn't bring the guests small portions of easily digestible food. Instead, the waiters serve the guests actual light. Next, when Azaz suggests a "square meal," the waiters don’t bring in a robust, balanced, hardy plate of food for everyone. Instead, the waiters give everyone squares. You could argue the extreme literalness of the banquet is a thing that has gone crazy in the absence of Rhyme and Reason.
Another thing that has seemed to go crazy due to the absence of Rhyme and Reason isn’t a thing but a person: it’s Mathemagician. You could say Mathemagician’s preoccupation with exact quantities defies common sense. You could also argue that the way in which he treats the rubies and gold as trash is foolish or “crazy.”
As for a third thing that has gone crazy, you might discuss the boy who doesn’t touch the ground or the soundless Valley of Sound.