Let me improve somewhat on Farmer's lower bound.

**Theorem.** If there is a cardinal $\kappa$ with the stated reflection property, then there are many measurable cardinals, measurable cardinals of very high Mitchell rank, and indeed, $1$-extendible cardinals of high Mitchell rank. 

The update gives a cardinal $\bar\kappa$ that is $\alpha$-extendible for $\alpha$ much larger than many measurable cardinals above $\bar\kappa$. In particular, $\bar\kappa$ is highly partially supercompact.

**Proof.** Suppose that $\kappa$ has the stated reflection property. Consider some large ordinal $\theta$ and let $B=\langle V_{\theta+1},{\in}\rangle$. By the reflection property, there is some structure $A$ with lots of elementary embeddings $j:A\to B$ hitting any desired target. We may assume $A$ is a transitive set. Let $\bar\kappa$ be the smallest critical point of such an embedding. For any $X\subseteq\bar\kappa$, it is in $B$ and so there is some $x\in A$ and $j:A\to B$ with $j(x)=X$. Since $x$ and $j(x)=X$ must agree up to $\bar\kappa$, this implies $X\in A$. So $P(\bar\kappa)\subseteq A$. This implies that $\bar\kappa$ is measurable, since we can define the induced normal measure $X\in\mu\iff \bar\kappa\in j(X)$ for such a $j$ with critical point $\bar\kappa$. 

So we've seen that $\bar\kappa$ is a measurable cardinal. And since $B$ also can see this, there must be measurable cardinals below $\bar\kappa$ in $A$. And indeed, the measure $\mu$ on $\bar\kappa$ will concentrate on measurables. So $\bar\kappa$ has Mitchell rank 1. But $B$ sees this, and so $\bar\kappa$ has Mitchell rank 2, and so forth, cycling around the loop for a long while to get very high Mitchell ranks.

The same idea shows that $\bar\kappa$ is 1-extendible, since we must have $V_{\bar\kappa+1}\subseteq A$ and so $j\upharpoonright V_{\bar\kappa+1}:V_{\bar\kappa+1}\to V_{j(\bar\kappa)+1}$ witnesses 1-extendibility. And since this is inside $B$, we get that $\bar\kappa$ is a limit of 1-extendibles, of high Mitchell rank again. $\Box$ 

If we could show $P(P(\bar\kappa))\subseteq A$, we would get $2$-extendibility, and so forth. 

**Update.** Let me now explain, based on the arguments in the comments by Farmer and Andreas, that we can achieve higher degrees of extendibility. The main idea is to show that $V_{\bar\kappa+\alpha}\subseteq A$ for higher ordinals $\alpha$, which shows that $\bar\kappa$ is $\alpha$-extendible. The argument I've given above shows how the case $\alpha=1$ works. Let's now extend further. 

Let $\bar\kappa_0=\bar\kappa$ be the least critical point that arises with $j:A\to B$, and let $\kappa_1$ be the next critical point. Such a critical point must arise because there must be $j:A\to B$ with $\bar\kappa$ in the range of $j$, and such an embedding must have critical point above $\bar\kappa$. What I claim is that $P(\bar\kappa_1)\subseteq A$. To see this, consider any $X\subseteq \bar\kappa_1$, and then find a $j:A\to B$ with $\{\bar\kappa,X\}\in\text{ran}(j)$. It follows that $\bar\kappa\in\text{ran}(j)$ and so the critical point of $j$ is at least $\bar\kappa_1$, and so $X\in A$ by the same reasoning as before. 

The main point is that the embeddings $j:A\to B$ with critical point $\bar\kappa$ now are witnesses of $\bar\kappa_1$-extendibility, which is already a high degree of extendibility. And we can continue with $\kappa_\alpha$ etc. getting very high degrees of extendibility of $\bar\kappa$ this way. It seems we get that $\bar\kappa$ is $\alpha$-extendible, where $\alpha$ is the $\bar\kappa$th measurable cardinal above $\bar\kappa$. And much more.

Andreas mentioned in the comments an idea for pushing this further, and perhaps he will post an answer about that.

**Another feature.** There is another certain feature here I noticed that seems interesting, and we might be able to push it much harder, but I don't quite see how to use it yet. Namely, for every $\theta$ we got a small transitive set $A$ which supports the elementary embeddings $j:A\to V_{\theta+1}$. Since there are only set many such $A$, it must be that some $A$ works for unboundedly many $\theta$. That is, we have a single transitive set $A$, such that for arbitrarily large $\theta$ we have elementary embeddings $j:A\to V_{\theta+1}$ that cover the target. And there will be a $\bar\kappa$ in $A$ that is the critical point of such an embedding for arbitrarily large $\theta$. That seems powerful, but I'm not sure exactly how to use it. It isn't quite super-$1$-extendibility, since perhaps it isn't $\bar\kappa$ that is sent high, even though the target model of $A$ can be made high.