It's not particularly natural, but I'm thinking the set of irrational numbers whose continued fraction expansion $[a_0; a_1, a_2, \ldots]$ is such that $[a_0; a_2, a_4, \ldots]$ is quadratic irrational should be an example.

Smooth manifolds? The OP said to work with additive categories.

Very often people post answers in comments when they are hesitant about whether a question is on-topic here: they don't necessarily want to encourage more such questions, but also they don't want to leave the OP empty-handed. Of course I'm not sure that's Jeremy's thinking, but just to let you know this sometimes happens.

I am one of those at MO who has complained about this proof. However, I was shown the light about what is really going on there by this pretty wonderful Mathologer video on youtube: youtube.com/watch?v=DjI1NICfjOk. (Oh, and now I see that Moritz Firsching explained the same thing in the other thread.)

(The last comment was in reply to a comment which has now been deleted.)

Start with the observation that a subobject of the minimal subobject $0$ of $1$ is $0$ again. Let $X$ be any object. Then there is a "singleton map" $\sigma: X \to PX$ which is monic, and there is also a map $1 \to PX$ which classifies the subobject $id_X: X \to X$. Take the pullback of the subobject $\sigma$ along $0 \to 1 \to PX$. This gives a subobject of $0$, which is $0$ again, and this gives a map $0 \to X$.

There's a theorem that an elementary topos has finite colimits. The usual construction (the theorem is that the power object functor $P: E^{op} \to E$ is monadic) is somewhat complicated though; details are in the book by Mac Lane-Moerdijk. A more elementary approach is to develop enough "internal logic" (conjunction, implication, and universal quantification), and then define the initial object as the internal intersection of all subobjects of 1. More on this in this paper: www2.mathematik.tu-darmstadt.de/~streicher/CTCL.pdf

Re Goldblatt: well, unfortunately not entirely. Maybe I'll spend some time thinking about your question.

Perhaps one should be aware that Szabo's Algebra of Proofs has a somewhat notorious reputation of having a lot of mistakes in it. (Let me add, for the sake of balance, that I found the book very useful in getting my bearings when I was doing dissertation work; however, I wouldn't take him as an authority on topos theory.)

I’m voting to close this question because the question was asked and answered at CSTheory.SE.