Add one place function $V$ to the language of set theory.
Add axioms of: Extensionality; Separation, stipulated in the usual manner.
Define $``ordinal"$ along Von Neumann's.
Stages: $\forall \ ordinal \ \alpha: V_\alpha = \{x \subseteq V_\beta | \beta < \alpha\}$
Foundation: $\forall x \exists \ ordinal \ \alpha \ ( x \in V_\alpha )$
Infinity: $\exists \alpha [\alpha \neq \emptyset \land \forall \beta < \alpha \exists \gamma (\beta < \gamma < \alpha)]$
Height: $\forall x \exists \ ordinal \ \alpha: \exists f \ (f: x \rightarrowtail \alpha) $
I personally consider this fragment of ZFC which appears to have $V_{\theta}$ [where $\theta$ is the first fixed point $\theta=\omega_\theta$] as its domain; to be the most natural of impredicative set theories, more natural than both ZC and ZFC
Now we can go beyond that easily to heights passing those of ZFC, for example we can add a single axiom stipulating that there exists a regular limit of regular cardinals, and this would be the first weakly inaccessible, and the theory would pass beyond ZFC.
Can we easily extend the above theory to just ZFC, and what I mean by "easy" here is adding a single axiom for that sake? In other words a single statement that make ordinals form only as long as they are accessible!
IF that cannot be done, then this means (in my own opinion) that ZFC is to some extent not a seamless embodiment of the notion of a particular hierarchy?!
Also I'd like to add this question since its directly connected to this subject:
Can we easily extend the above theory to some extension of ZFC?