This is, alas, in large part a series of questions on unpublished work of Hugh Woodin; it's also quite frivolous if Reinhardt cardinals turn out inconsistent.
Definitions:
Call $\kappa$ an $I-1(\kappa,\delta)$ cardinal if it is the least ordinal lifted by a nontrivial elementary embedding $j:V_{\delta+2} \to V_{\delta+2}$ where $\delta$ is an uncountable strong limit cardinal of countable cofinality and the supremum of the critical sequence $\kappa_n$.
Call $\kappa$ a weak Reinhardt cardinal if it is $I-1(\kappa,\delta)$, there is an ordinal $\gamma \gt \delta$, and elementary substructures $V_{\kappa} \prec V_{\delta} \prec V_{\gamma}$.
Call $\kappa$ a Reinhardt cardinal if it is the critical point of $j:V \to V$.
Call $ZFR$ the base theory $ZF$ in the language $(\in,j)$ augmented with a scheme asserting $\{\exists\kappa\mid \kappa\text{ is Reinhardt}\}$
Question 1: What does $ZFR$ imply?
Woodin folklore mentions a proof that $ZFR$ implies the existence of models of all large cardinal axioms not known to refute Choice. What is the proof $ZFR \implies$ "There exists a transitive model of $ZFC + I1(\kappa,\delta)$", and why does it not immediately extend to a proof of "There exists a transitive model of $ZF + DC_{\delta} + I-1(\kappa,\delta)$"?
Reinhardt cardinals (trivially) satisfy any large cardinal property derived from $j:V \to M$ and so are themselves at least $I2$ in some sense, but I don't know how to proceed further, and Woodin frequently mentions that there is no known proof of the expected implication Reinhardt $\implies$ weak Reinhardt. And while David Asperò has recently published in A short note on very large large cardinals a nice proof that Reinhardt cardinals imply many $j:V_{\mu} \to V_{\mu}$ with the target of the embedding moved arbitrarily high, these $\mu$ are poorly constrained and $V_{\mu}$ in general satisfies neither Replacement nor Choice. They may, therefore, not even be rank-into-rank in the usual sense.
Question 2: What's "next" after $ZFR$?
The rare tome On Woodin's Investigations into "Here there be Dragons" contains a proof that $j:HOD \to HOD$ follows from a strong extension of $ZFR$. What is this extension?
The strongest such axiom I recall encountering in the literature is a scheme asserting that there is an elementary substructure $V_{\kappa} \prec V$. However perilously strong, this axiom is not super $n$-helpful for determining what sort of cardinal structure the Reflecting Reinhardt cardinal is duplicating. Thus I'm hoping the answer might be something else, more useful for establishing answers to Question 1.
Question 3: What is the point?
Do these very large large cardinals hint at whether the Axiom of Choice constrains the consistency strength of $ZFC$, or not?
Here we arrive at the soft and murky motivation behind these questions. In light of Kunen's Inconsistency Theorem, it's possible to ask if the Axiom of Choice is close kin to the Axiom of Constructibility, something that places a sharp bound on the amount of consistency strength $ZF$ can support. Under this interpretation, if $I-1(\kappa,\delta)$ is consistent with $ZF$ then there is no equiconsistent large cardinal axiom consistent with $ZFC$. And although there are various shenanigans we can perform to sneak the strength back in (e.g. work in $ZFC +$ Con($ZFR$)), these don't tell us much about what $V$ is like. Further, they raise the question of why we oughtn't just work in $ZFC + V=L$ and get our measurables from some $j:M \to N$, since the Axiom of Constructibility is doing the same job as Choice in this context, only better.
Contrariwise, it's also possible to ask if it's the definability restriction of $V=L$ that renders it so inhospitable to large cardinals, and to assume that if a prospective axiom is too strong for the notoriously non-constructive Axiom of Choice, then it was outright inconsistent to begin with and Choice has merely facilitated the proof. Woodin's $HOD$ Conjecture is work in this vein (it refutes $I-1(\kappa,\delta)$), and even if the $HOD$ Conjecture is false, $I-1(\kappa,\delta)$ might be inconsistent for some currently unknown reason.
The sort of relevant evidence I'm hunting for is a large cardinal property or forcing axiom that:
1. is very strong in ZFC
2. is weaker but not trivial in ZF
3. indicates that some property a Reinhardt cardinal should have has leaked away through the absent wellordering of some $P(\aleph_{\alpha})$
The first two criteria are satisfied by a scant few things I can think of, including for example some generalizations of Chang's Conjecture beyond $(\omega_{3},\omega_{2}) \to (\omega_{2},\omega_{1})$; these generalizations, however, either have a cardinal structure that doesn't illuminate the third requirement or they have a cardinal structure that I find baffling and unwieldy. (And yes, that third requirement is so vague that the question may not admit an answer even if an example exists. If I knew how to make it precisely state some such property I would, but in general we don't know what those properties are!)
First post to this site; no snub intended if I don't respond in comments =)
Edit:
Hello Joel, and thank you for the warm welcome. For $ZFR$ I wanted
1. All the Axioms of $ZF$ in the language $\{\in\}$
2. Replacement for formulae in the language $\{j,\in\}$
3. An Axiom scheme asserting $j:\langle V,\in \rangle \to \langle V,\in \rangle$
…and not that there are many critical points. This is, however, probably a mistake (or at least overly specific) and I'll gladly substitute any formalization of "$ZF$ + There is a Reinhardt cardinal" that doesn't force the embedding, by elementarity, to be the identity, even if that formalization is implicit and be understood in a fully two-sorted metatheory. Particularly if that's what Woodin had in mind for his proofs.
And yes, the first two requirements should easy enough to formalize (though I also would like $ZF + \psi \vDash$ Con($ZF$) $\wedge$ $ZFC + \varphi \vDash$ Con($ZF + \psi$) at the very least). It's the third requirement that I don't know how to formalize without an applicable, rigorous notion of "large cardinal property". Woodin's won't suffice in full generality when I don't know how to express a given property as some $j:V \to M$ with appropriate constraints on $M$. And in the case of Reinhardt cardinals, I'm not even sure which properties to expect.
Corazza's Wholeness Axiom is almost exactly what I'm looking for, thank you for the reminder. The failure of Replacement for $j$-classes acts as a pressure valve, disallowing the critical sequence from forming a set, and thereby preventing the absurdity of cof$(Ord) = \omega$ and reducing the consistency strength of Reinhardt cardinals to something weaker than $I3(\kappa,\delta)$ and stronger than a cardinal super $n$-huge for all $n \in \omega$, which is quite manageable. Unfortunately, it rules out the same hugeness that Kunen's theorem does, and doesn't supply new information about how a fully Reinhardt cardinal must appear.
I think? It would be nice to be wrong about that! And you've studied it more than most.
Best Answer
The following is due to Woodin:
Theorem. Assume $ZF$+ there exists a Reinhardt cardinal + there exists a proper class of supercompact cardinals is consistent. Then there exists a genric extension of the universe which satisfies $ZF$ + the axiom of choice + there exists a proper class of supercompact cardinals, and such that in it Woodin's $HOD$ conjecture fails.