The word "perfidy" means to be untrustworthy or deceitful. Merriam-Webster online dictionary defines perfidy as being disloyal or unfaithful and it is a synonym for treachery. 

Although chapter 13 is called "Perfidy Unlimited," the word was first used in chapter 8 when Despereaux's father reported his son to the Mouse Council. Ostensibly, "perfidy" in chapter 13 refers mainly to Furlough, the brother of Despereaux, and his betrayal. However, the word applies to multiple characters since its introduction in chapter 8. 

The Most Very Honored Head Mouse says that Despereaux "cannot be trusted" because of his interactions with humans, directly invoking the concept of treachery that soon the entire mouse community will engage in against Despereaux. Several pages later in chapter 9, Furlough says to himself, "I was right to turn him in. He's sick." And in chapter 12 Despereaux's mother shows a great deal of concern for her son, yet refuses to stand up for him or to say anything truly of comfort. Instead, all she says is "adieu."

This all leads up to chapter 13's title, "Perfidy Unlimited," stating quite plainly that Despereaux has been betrayed infinitely. Furlough being one of the mice to lead him to the dungeon is the last and perhaps the heaviest ounce of perfidy to befall Despereaux.