Just some minor points: 1. The situation with the Axiom of Regularity (well-foundedness of the $\in$-relation) is similar to the situation with the Axiom of Choice. In any model of ZF without regularity we can build the usual von Neumann hierarchy $V_\alpha$, $\alpha$ an ordinal, by iterating the power set operation, and the union of the $V_\alpha$'s is an inner model that satisfies regularity. 2. Depending on the precise formulation, the Replacement Scheme often implies the Separation Scheme. 3. As you indicate, Infinity is indispensable, since $V_\omega$ is a model of all the other axioms, and even if the background theory is full ZFC (i.e., if we pretend to live in a universe satisfying ZFC), from $V_\omega$ we cannot build a model of ZFC. 4. I have to confess that I am not sure what you mean by the Abstraction Scheme. Is this what I would call Separation? Anyhow, given an uncountable regular cardinal $\kappa$ (such as $\aleph_1$), $H_\kappa$, the collection of all sets whose transitive closure is of size $<\kappa$, is a model of ZFC without the Power Set Axiom (I am again assuming ZFC as my background theory). It follows that there is no procedure to build a model of ZFC from a model of ZFC without Power Set. So it seems that any minimal subsystem of ZFC would have to include the Power Set Axiom. Partial Conclusion so far: You need infinitely many axioms by Joel's answer, you don't need AC or Regularity, but you do need Infinity and Power Set.