What makes "the topos $\mathbf{M}_2$" such a good counterexample? I'd like to ask this question sooner rather than later; it might be a bit half-baked. So I'm sorry. It's just that there's a chance I'll be side-tracked from Topos Theory for a couple of months (with some group-theoretic ideas).

What makes "the topos $\mathbf{M}_2$" such a good topos-theoretic counterexample?

Goldblatt makes heavy use of $\mathbf{M}_2$-$\mathbf{Set}$, referred to as "the topos $\mathbf{M}_2$," as a source of counterexamples to various phenomena in Topos Theory; in fact, he calls it the "canonical" and "universal" counterexample.

Is this topos unique in its pathology?

Let's have a recap of the definitions:
Definition 1: The monoid $\mathbf{M}_2$ is given by $(2=\{0,1\}, \cdot, 1)$, where $\cdot$ is defined by $$1\cdot1=1,\quad\quad 1\cdot0=0\cdot1=0\cdot 0=0.$$
Definition 2: An $\mathbf{M}_2$-set is a pair $(X, \lambda)$, where $X$ is a set and $\lambda: \mathbf{M}_2\times X\to X$ is an action of $\mathbf{M}_2$ on $X$.
Definition 3: The topos $\mathbf{M}_2$-$\mathbf{Set}$ is the category whose objects are $\mathbf{M}_2$-sets and whose morphisms are action-preserving functions. A proof that it's a topos is given here (with $M$ as $\mathbf{M}_2$).

Specific examples of how this thing acts as a counterexample are currently beyond my ability to explain. They're several chapters deep into Goldblatt's book.

Thoughts: I don't have anything non-trivial to say. (I've already given the basic definitions so I don't want to insult your intelligence . . . ).
 A: Note that a set $X$ with an $\mathbf{M}_2$-action (or, as I usually think of it, an $\mathbb{F}_1$-action) is the same a surjective map $0\cdot - : X=X_1\to X_0$, together with a section $X_0 \to X_1$ (which I'll treat as an inclusion).
This correspondence also gives us a (functorial) relationship between $\mathbf{Set}$ and $\mathbf{M}_2$ as topoi (which I don't have time to work out at the moment, but should be somewhat straightforward).
If we think of $\mathbf{Set}$ as the logic of vanilla set theory, we can think of $\mathbf{M}_2$ in the following way.  Say that a proposition can be "really true" or "kind of true".  We have the following logic: "really really true = really true", "really kind of true = kind of true", et cetera.  For example, if we want to test whether "$x\in X$" is true—then we say that it's "really true" if $x\in X_1\setminus X_0$, but only "kind of true" if $x\in X_0$.
So, in some sense, the language of $\mathbf{M}_2$ is a minimal enrichment of standard Zermelo-Fraenkel theory that allows for this semantics of fuzziness.  As such, it is a very reasonable thing to think of it as a universal example of a set theory without the law of the excluded middle, but it is hard to say more without looking closely at what Goldblatt intends to say here.
