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We study nilpotency in the context of exact Mal'tsev categories takingcentral extensions as the primitive notion. This yields a nilpotency towerwhich is analysed from the perspective of Goodwillie's functor calculus. Weshow in particular that the reflection into the subcategory of $n$-nilpotentobjects is the universal endofunctor of degree $n$ if and only if every$n$-nilpotent object is $n$-folded. In the special context of a semi-abeliancategory, an object is $n$-folded precisely when its Higgins commutator oflength $n+1$ vanishes.

Central reflections and nilpotency in exact Mal'tsev categories