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.
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 $j(X)$ must agree up to $\bar\kappa$, this implies $X\in A$. So $P(\bar\kappa)\subseteq A$ and consequently also $H_{\bar\kappa^+}\subseteq A$. This implies that $\bar\kappa$ is measurable, since we can use the induced normal measure.
This implies that $\bar\kappa$ is a measurable cardinal. And since $B$ can see this, there must be measurable cardinals below $\bar\kappa$ in $A$. And indeed, the measure on $\bar\kappa$ will concentrate on measurables. So $\bar\kappa$ has Mitchell rank 1. And thus also Mitchell rank 2 and so forth, for a long while.
The same idea shows that $\bar\kappa$ is 1-extendible, since we 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. $\Box$
Next steps. I noticed something that might enable one to push things much harder, but I don't quite see how to use it. Namely, for every $\theta$ we get 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.