The question is extremely dependent on how size issues are handled, and the presentation given inmany choice that can be made, so that it is very hard to give a general answer.
If you really work with general presheaves categories then I cannot think of any interesting functors $C_i \to C_{i+1}$ that would define an interesting value at limit ordinal. For example the post doesn't make sense on its own:Yoneda embedding is not defined as if $C_0 \neq \varnothing$, $C_1$ will be a large category, so $C_{2}$ will have large hom-setsets, and so the Yoneda embeddings $C_2 \to C_3$ will not be defined unless "Set" also become bigger as the process go.., in which case the process will obviously never stabilize.
Also if you allowSimilarly, the covariant functoriality of the presheaf construction as suggested by Peter Lumsdain in the comment is not available either as it is no longer defined when working with large setscategory so there are no way to use it. The contravariant functoriality is available always, you also needbut I do not see how to consider large ordinalsuse to construct a tower.
In any caseTaking the process will not stabilize atcolimit to be a disjoint union $\omega$( colimit over a discrete category) is not a viable solution either: the presheaf category are always connected categories - in fact(they have initial objects and terminal objects for most way to handleexample) while the size problem it won't stabilize at allcoproduct are always disconnected, but here is one version of it where it does stabilizeso the value at a limit ordinal and at a successor ordinal will never agree.
To avoid havingThe only way I can think of to increase "Set" at each stepmake sense of this construction, I'm replacingis to replace the category $Set^{C^{op}}$ by the construction $\widehat{C}$ the free co-completion of $C$ under all. When $C$ is small colimitsthen $\widehat{C} \simeq Set^{C^{op}}$, thatwhen $C$ is itslocally small then $\widehat{C}$ the full subcategory of $\widehat{C} \subset Set^{C^{op}}$$Set^{C^{op}}$ of so called "small presheaves", that is the presheaves that are small colimitcolimits of representable functors (If. When $C$ has a large hom-set then $\widehat{C}$ is no longer related to $Set^{C^{op}}$, but the advantage of iterating $\widehat{C}$ is that one never gets out of locally small it doesn't change anything)categories.
Here To fix what "small" means, I chose an inaccessible cardinal $\kappa$ (regular would be enough in fact), and "small" means $\kappa$-small. Interestingly, $\kappa$ doesn't need to be inaccessible, regular is enough.
ThenOnce we restrict to small presheaf, then there are at least two interesting way to build such tower. One can use the Yoneda embeddings, or one can as suggested by Peter Lumsdaine, or one can start with some functor $C_0 \to C_1$ and then use the co-variant functionality of $\widehat{C}$ to build functor $C_1 \to C_2$ etc...
I claim that the process "stops" at the ordinal $\kappa$, in the sense that $C_\kappa \simeq C_{\kappa+1}$ and notit never stops before. (so if you only induce on small ordinal it never stops).
In general,If one use Peter Lumsdaine's version then the process also clearly stops at the ordinal $\kappa$ (and this time really stay at the value it had for $\kappa$). I haven't thought to this specific example enough, but it is likely that one can imagine a few versions ofshow that it never stabilize before $\kappa$.
If one really wants to keep the construction $Fun(C^{op}, Set)$ without restricting to small presheaves, and one somehow find a way to make it into an interesting tower (which I don't think the processis possible), then it will ever stop before the first non-small ordinalvery probably never converge:
Indeed, and things might be much worse if you increase "Set" as you goit is for example easy to show that there is no category $C$ such that $Fun(C,Set) \simeq C$ using several notions(a 2-categorical version of smallness) Lawvere fix-point theorem and the fact that $Set$ has some endofunctors without no fix-point (for example the covariant power-set endofunctor).
I suspect one can adjust this argument to show that there is no category $C$ such that $C \simeq Fun(C^{op},Set)$ as well, but I can't quite find the argument right now. This would clearly show that no variant of the construction with $Fun(C^{op},Set)$ would ever converge.
