**Question:** Is it consistent (relative to the existence of large cardinals) that there is an elementary embedding $j\colon V\to M$ (where $M$ is transitive model) that factors as $j = j\_n \circ k\_n$ for $n < \omega$, such that for every $n$, $\text{crit }j\_n = \kappa$ (the same ordinal), but $j\_{n} (\kappa) < j\_{n + 1} (\kappa)$ for every $n$?


The motivation for this question is the following observation: It is possible to get from single measurable cardinal many different elementary embeddings $j\_n\colon V \to M\_n$ with the same critical point (by using iterated ultrapower) and a single model $M$ and elementary embedding $j \colon V \to M$ such that $j = k\_n \circ j\_n$ for $k\_n \colon M\_n \to M$. The question is whether the "dual" situation, which seems to be more problematic, is possible.



Elementary embeddings with the same critical point
Yes, this situation can occur. One should simply undertake the dual of the construction you had suggested with iterated ultrapowers. 


Specifically, suppose that $\mu$ is a normal measure on a measurable cardinal
$\kappa$ and let $j:V\to M\_\omega$ be the embedding arising from
iterating the ultrapower $\omega$ many times. Thus, $M\_\omega$ is
the direct limit of the system of embeddings $j\_{n,k}:M\_n\to M\_k$,
where $j\_{n,n+1}$ is the ultrapower of $M\_n$ by
$\mu\_n=j\_{0,n}(\mu)$. The sequence $\langle\kappa\_n\mid
n<\omega\rangle$ is the critical sequence.


For any set $S\subset\{\kappa\_n\mid n<\omega\}$, we may form the
*seed hull* $$X\_S=\{j(f)(\vec s)\mid f:\kappa^{<\omega}\to V, f\in
V, \vec s\in S^{<\omega}\},$$
and this is an elementary substructure of $M$,
which can be seen by verifying the Tarski-Vaught criterion, and it contains the range of $j$. Further, it is a basic fact of normal ultrapowers that no seed
$\kappa\_n$ can be generated from the others. That is, if
$\kappa\_n\notin S$, then $\kappa\_n\notin X\_S$. You can find this and related results in section 3 of my paper, [Canonical seeds and Prikry trees](https://jdh.hamkins.org/seeds/), JSL 62, 1997 (adapted from a chapter of my dissertation). It uses the
normality of $\mu$, and this particular fact is not necessarily true without that
assumption, as shown in the paper. 


Now, for each finite $n$, let $S\_n=\{\kappa\_m\mid m\geq n\}$ and
let $\pi\_n:X\_{S\_n}\cong N\_n$ be the Mostowski collapse of
$X\_{S\_n}$. Thus, rather than including all seeds up to $\kappa\_n$, which is how you might have proved the situation you mentioned at the end of your question, we instead undertake the dual set, using only seeds from $\kappa\_n$ and upward. Let $k\_n=\pi\_n\circ j:V\to N\_n$, and let
$j\_n=\pi\_n^{-1}:N\_n\to M$, so that we have a commutative triangle
of elementary embeddings $j=j\_n\circ k\_n$.


Since $S\_n$ contains only $\kappa\_m$ for $m\geq n$, it follows
that $\kappa\_i\notin X\_{S\_n}$ for $i<n$, and consequently $X\_{S\_n}\cap[\kappa,\kappa\_n)=\emptyset$, leading to 
$\pi(\kappa\_n)=\kappa$. Thus, $j\_n(\kappa)=\kappa\_n$. In
particular, these all have the same critical point, and they reach
higher as $n$ increases. So the situation is just what you
requested.


