Let X be a set. Which functions from X to itself cannot be non-decreasing if we equip X with any well-order ≤, the same in X as the domain and X as the codomain?

Such functions obviously exist. For example, if we take X={0,1} and the function f(x)=~x ("~" denoting negation), then there is no well-order on X making this a non-decreasing function.

But what are these functions in general? Does a composition of such functions also have this property?