Examples of collapsing hierarchies Are there interesting examples of &quot;collapsing hierarchies&quot; in computer science?

The formal definition of a hierarchy here would be a class of languages/problems/objects which is parametrized by a partially ordered set. But I am of course looking for an at least somewhat &quot;natural&quot; hierachy, where the parameter represents some type of resource. I am particularly interested in hierarchies where the parameter space is $\mathbb{N}$ or the $\Sigma_i/\Pi_i$ ladder. Some examples are

the polynomial hierarchy (counting alternations),
the lightface hierarchy (same),
$NC^i$ and $AC^i$,
various ways to associate languages to $k$-headed finite-state automata (as $k$ grows).

I am also happy with more complicated parameter spaces such as polynomial time languages parametrized by their time-complexity or space-complexity, if you know interesting collapsing results for them.
By collapsing I simply mean that some level of the hierarchy already contains all elements of the hierarchy. In all natural cases I can think of, either we know hierarchy does not collapse or we don't know whether it collapses. I would like to know some examples of hierarchies that do collapse. The hierarchy should be &quot;natural&quot;, and I would prefer an example where the collapse does not happen &quot;on the first level&quot;.
I did not carefully check that none of the hierarchies listed in What hierarchies and/or hierarchy theorems do you know? collapse, but at least it is not known to collapse for any of the ones I was familiar with, or for which this was mentioned in the answers.
The analogue of the $\mathsf{NC}$ hierarchy for algebraic circuits is known to collapse to the second level. That is, algebraic circuits of size $n^{O(1)}$ computing a polynomial of degree $n^{O(1)}$ can be rebalanced to have depth $O(\log^2 n)$ while only increasing the size by a polynomial factor. This is due to Valiant, Skyum, Berkowitz, and Rackoff. It is open whether $\mathsf{NC}^1 \stackrel{?}{=} \mathsf{NC}^2$ in this setting, but I believe the popular opinion is that $\mathsf{NC}^1 \neq \mathsf{NC}^2$ here.
If one further restricts to syntactically multilinear algebraic circuits, then we in fact know $\mathsf{NC}^0 \subsetneq \mathsf{NC}^1 \subsetneq \mathsf{NC}^2 = \mathsf{NC}$ unconditionally. The fact that $\mathsf{NC}^1 \subsetneq \mathsf{NC}^2$ is due to Raz, and the fact that $\mathsf{NC}^2 = \mathsf{NC}$ is due to Raz and Yehudayoff.
Raz, Ran, Separation of multilinear circuit and formula size, Theory Comput. 2, Paper No. 6, 121-135 (2006). ZBL1213.68301.
Raz, Ran; Yehudayoff, Amir, Balancing syntactically multilinear arithmetic circuits, Comput. Complexity 17, No. 4, 515-535 (2008). ZBL1188.68367.
Valiant, L. G.; Skyum, S.; Berkowitz, S.; Rackoff, C., Fast parallel computation of polynomials using few processors, SIAM J. Comput. 12, 641-644 (1983). ZBL0524.68028.