We introduce the concept of an $n$-inflation of a semigroup. In particular, for $n=1$ we obtain the inflation introduced by Clifford [6], and for $n=2$ the strong inflation introduced by Petrich [10]. We also characterize $n$-inflations of unions of groups, of semilattices of groups of unions of periodic groups, etc. In addition, we describe nilpotent semigroups of arbitrary nilpotency class.