By Gödel's incompleteness theorem, we can't prove ZFC consistent in ZFC. But "naturally" we believe so. So we could add the axiom "ZFC is consistent" and call the new axiom set ZFC-1. We could then define ZFC-α for any ordinal α (or any countable ordinal if that matters), consisting of all axioms in the previous ZFCs, and the axiom that all previous ZFCs are consistent. "Naturally", we still believe such axiom sets consistent.
But is this really a reasonable statement? My question is, where exactly does it stop? Even if we say they are all true, i.e. all the extensions of ZFC are consistent, by Gödel's incompleteness theorem, we would still be able to make higher levels, and involve some discussions about how proper classes are different from sets. But is it possible that we would already have to stop much before that? Say, could there be a big ordinal α (countable if that matters) not provable or even describable in ZFC, or having other unfavorable characteristics, making ZFC-α not properly definable, and different extensions to ZFC to make the beliefs intelligible have different interpretations about their exact meaning? Or conversely, could we know all such beliefs, though not provable and possibly indescribable, exist and have exact meaning with some reasonable assumptions?
Since there are only countably many recursive axiom systems, the axiom system must become nonrecursive at some countable ordinal and then you can't add a new consistency statement.