The question is extremely dependent on how size issues are handled, and the presentation given in the post doesn't make sense on its own: $C_1$ will be a large category, so $C_{1}$ will have large home set, and so the Yoneda embeddings $C_i \to C_2$ will not be defined unless "Set" also become bigger as the process go...

Also if you allow large sets, you also need to consider large ordinals.

In any case the process will not stabilize at $\omega$ - in fact for most way to handle the size problem it won't stabilize at all, but here is one version of it where it does stabilize.


To avoid having to increase "Set" at each step, I'm replacing the category $Set^{C^{op}}$ by $\widehat{C}$ the free co-completion of $C$ under all small colimits, that is its full subcategory $\widehat{C} \subset Set^{C^{op}}$ of presheaves that are small colimit of representable functors (If $C$ is small it doesn't change anything). To fix what "small" means, I chose an inaccessible cardinal $\kappa$ (regular would be enough in fact) and "small" means $\kappa$-small. 


Then I claim that the process stops at the ordinal $\kappa$ and not before. (so if you only induce on small ordinal it never stops).


In general, I can imagine a few versions of the construction, and I don't think the process will ever stop before the first non-small ordinal, and things might be much worse if you increase "Set" as you go using several notions of smallness...