 Nested partial unary operation category

An object of ${\mathcal{C}}_n$ is a set with a finite list of partial unary operations. The first is total; the domain of each later operation consists exactly of the defined fixed points of the immediately preceding operation. Morphisms preserve every defined operation. Take ${\mathcal{C}}_0$ to be the Category of sets. This tower gives examples of arbitrary finite monadic length even though its long free–forgetful composites induce the same first monad.