What is the consistency strength of a cardinal $\kappa$, such that there is some $j: V\prec M$ such that $M^{\lt j^\omega(\kappa)}\subseteq M$; in other words, for every cardinal $\lambda\lt\delta$, $M^\lambda\subseteq M$, where $\delta$ is the least fixed point of $j$. Let's call such a cardinal **almost $\omega$-huge**.

It is a known theorem of $ZFC$ that there can be no $\omega$-huge cardinal; i.e. $M^\delta\nsubseteq M$. However, there seems to be no known information about almost $\omega$-huge cardinals. To my knowledge, the standard methods of deriving contradiction do not work in this scenario.

$V_{\delta+2}$ need not be a subset of $M$, so $j\restriction V_{\delta+2}$ need not be an elementary embedding. Furthermore, $|j"\delta|=\delta$, so $j"\delta$ need not be in $M$. To my knowledge, there has been no research on this front. So what is the consistency strength of an almost $\omega$-huge cardinal?

**Results**

Every almost $\omega$-huge cardinal is $I2$.

Every almost $\omega$-huge cardinal is a limit of $I2$ cardinals, because the sequence of ultrafilers witnessing $I2$ is in $M$.

$j"\delta\notin M$ and $V_{\delta+1}\notin M$.

My hunch is that this is weaker or equivalent to $I1$ cardinals, but I have no way to prove this.

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