Does the process of 'constructing the category of presheaves' always/never stabilize? Does it stabilize for some special class of categories?

That is, work in a foundation that allows for multiple levels of 'categorical largeness' and let $\mathcal{C}$ be a category. Recursively define

1. $\mathcal{C}_0=\mathcal{C}$,
2. $\mathcal{C}_{\alpha+1}={\bf Set}^{\mathcal{C}_\alpha^{op}}$,
3. $\mathcal{C}_\lambda=\varinjlim_{\alpha<\lambda}\mathcal{C}_\alpha$.

Does there always/never exist some $\alpha\in O_n$ such that $$\alpha\leq\beta\implies \mathcal{C}_\alpha\simeq\mathcal{C}_\beta?$$

I tried playing around with the case $\mathcal{C}={\bf Set}$ and it seems like we don't have the above property for finite $\alpha$, since ${\bf Set}$ is locally small and ${\bf Set}^{{\bf Set}^{op}}$ is not locally small (natural transformations are proper-class sized collections of sets), and higher finite iterations begin sending proper classes to sets etc. -- I don't know if this process stabilizes at $\omega$ or some higher limit ordinal, however.

Further, if we begin with the terminal category ${\bf 1}$ then ${\bf Set}^{{\bf 1}^{op}}\cong{\bf Set}$ so the above sequence repeats itself shifted back one step, and if we begin with the initial category ${\bf 0}$ then ${\bf Set}^{{\bf 0}^{op}}\cong{\bf 1}$, so the above sequence repeats itself shifted back two steps.