Recall the notion of locally presentable category (nLab):
Definition: Fix a regular cardinal $\kappa$; a set is $\kappa$-small if its cardinality is strictly less than $\kappa$. A $\kappa$-directed category is a poset in which every $\kappa$-small set has an upper bound. A $\kappa$-directed colimit is the colimit of a diagram for which the indexing category $\kappa$-directed. An object $a$ of a category is $\kappa$-small if $\Hom(a,-)$ preserves $\kappa$-directed colimits. A category is $\kappa$-locally presentable if it is (locally small and) cocomplete and there exists a set of objects, all of which are $\kappa$-small, such that the cocompletion of (the full subcategory on) this set in the category is the entire category.
It is a fact that every $\kappa$-locally presentable category is also $\lambda$-locally presentable for every
$\lambda > \kappa$.
In a current research project, we have some constructions that work naturally for $\kappa$-locally presentable categories for arbitrary regular cardinals $\kappa$. But all of our applications seem to be to $\aleph_0$-locally presentable categories. For example, for any ring $R$, I'm pretty sure that the category of $R$-modules is
$\aleph_0$-locally presentable. (Every module has a presentation; any particular element or equation in the module is determined by some finite subpresentation.) The category of groups is
$\aleph_0$-locally presentable (by the same argument). I'm told that every topos is locally presentable for some $\kappa$; is a topos necessarily
Indeed, although I know of some categories that are not
$\aleph_0$-locally presentable but are $\kappa$-locally presentable for some larger $\kappa$; the example, apparently, is the poset of ordinals strictly less than $\kappa$ for some large regular cardinal $\kappa$. But this is not a category I have ever encountered "in nature"; it's more of a zoo specimen. Hence my somewhat ill-defined question:
Question: Do there exist "in nature" (or, "used by working mathematicians") categories that are $\kappa$-locally presentable for some
$\kappa > \aleph_0$but that are not
Put another way, is there any use to having a construction that works for all $\kappa$?
So far, I haven't thought much about general topoi, and we do want our project to include those, so I would certainly accept as an answer "yes, this particular topos". But there might be other "representation theoretic" categories, or other things.