A list of locally finitely presentable topoi that are not coherent Coherent topoi play an important role in topos theory, especially in the interaction with logic. Their most handy characterization is provided by   Johnstone. Sketches, D3.3.1. Every coherent topos is locally finitely presentable (Johnstone. Sketches, D3.3.12), but the converse is not true.
Since I am not aware of many counterexamples to the converse, I would like to make a list of them. Possibly, I encourage you to name examples that pop up in nature, instead of something designed to answer the question.

Let me list a couple of them, both in Sketches, D3.3.12.

*

*$\text{G}$-$\mathsf{Set}$, for G an infinite group.

*$\mathsf{Set}^{{\text{Fin}}_{\twoheadrightarrow}}$, where ${\text{Fin}}_{\twoheadrightarrow}$ is the category of finite sets and surjections.


 A: Claim: In a locally finitely-presentable topos, the quasicompact objects can be characterized as the quotients of finitely-presentable objects.
Proof: Suppose first that $X$ is quasicompact. By local finite presentability, there is an effective epimorphism $\amalg_{i \in I} F_i \twoheadrightarrow X$ where each $F_i$ is finitely-presentable. Because $X$ is quasicompact, there is a finite subset $n \subseteq I$ such that $\amalg_{i \in n} F_i \twoheadrightarrow X$ is an effective epimorphism, so that $X$ is a quotient of a finitely-presentable object.
For the other direction, first note that quasicompact objects are closed under quotients. So it suffices to show that every finitely-presentable object $X$ is quasicompact. Let $\amalg_{i \in I} Y_i \twoheadrightarrow X$ be a cover. Then $X = \varinjlim_{n \subseteq I} X_n$ where $n \subseteq I$ is finite and $X_n$ is the image of $\amalg_{i \in n} Y_i$. Because $X$ is finitely-presentable, it is a retract of some $X_n$. Since $X_n \rightarrowtail X$ is monic, this means that $X = X_n$. So $\amalg_{i \in n} Y_i$ is a finite subcover of $X$, and $X$ is quasicompact.
Fact: (cf. Makkai and Reyes Thm 9.2.2) In a coherent topos, the coherent objects (which are in particular quasicompact) form a pretopos.
Upshot: One easy obstruction is that in a coherent topos, the terminal object needs to be a quotient of finitely-presentable objects.
Example: If $P$ is a poset which does not have a cofinal finite subset, then the presheaf category $Psh(P)$ is locally finitely-presentable but not coherent.
A: One important example of a non-coherent topos to have in mind is the presheaf topos $Set^{\mathcal F}$ where $\mathcal F$ is the category of finitely-presentable objects in the category of fields.
This example is an illustrative one, because the category of points of $Set^{\mathcal F}$ is the category of fields, so it's tempting to call it the "classifying topos for fields". But this is generally thought to be misleading, basically because $Set^{\mathcal F}$ is not coherent.
There is a coherent theory whose models in $Set$ are precisely the fields. It's called the "coherent theory of fields", and if you want to talk about "field objects" in a topos, you will typically use this theory / its classifying topos. But the classifying topos of the coherent theory of fields is not a presheaf topos.
I think this is discussed somewhere in the Elephant, and also in Beke's Theories of Presheaf Type.
