This is a copy of Math StackExchange question #4395977, which I felt was more appropriate for MathOverflow.
Note on terminology: "admissible", "$(^+)$-stable", and "$\Pi^1_1$-reflecting" can all be found in Madore's "A Zoo of Ordinals". $\Pi^1_1$-reflection is defined in Richter and Aczel's "Inductive Definitions and Reflecting Properties of Admissible Ordinals". The set theory $\mathrm{KP}+\Pi_1$-collection is defined in Arai's paper analyzing it.
The admissible ordinals $\alpha$ s.t. the supremum of $\alpha$-recursive well-orderings is less than the next admissible $>\alpha$ are called non-Gandy, and according to Madore's "A Zoo of Ordinals" the least non-Gandy ordinal is between the least $(^+)$-stable and the least inaccessibly-stable in size. Another phenomenon that begins happening with admissible ordinals at a similar size is iterated $\Pi^1_1$-reflection, which according to Taranovsky's "Ordinal Notation" page (cf. MO question #118854) can be informally iterated further than first-order reflection.
On the same page in §4.2, Taranovsky conjectures that even with an arbitrary recursive base notation O, his ordinal notation system "Degrees of Reflection" only just suffices to capture the structure below the least non-Gandy ordinal, in which case the structure of admissible ordinals below the least non-Gandy is quite rich. Also, from the same site's §5.2, an informal statement about how rich the structure of non-Gandy ordinals is even below the least $\alpha$ that's $(\alpha^++1)$-stable, sharpening the "inaccessibly-stable" bound from A Zoo of Ordinals, and that said structure is rich to the point of stratifying into multiple "levels" the same way the $\beta$-stable ordinals would for $\beta$ between $\alpha$ and the next recursively inaccessible $>\alpha$. And not only that, but §4.2 also says each $(\alpha^++1)$-stable $\alpha$ is non-Gandy, implying the non-Gandy ordinals are unbounded in the least inaccessibly-stable. All things considered, iterated $\Pi^1_1$-reflection and non-Gandy ordinals seem to contribute very heavily to the structure of countable admissibles past the least $(^+)$-stable!
However, in strong ordinal notations going past the level of the least $(^+)$-stable, such as the ones in Arai's recent analysis of $\textrm{KP}+\Pi_1$-collection ("An ordinal analysis of $\Pi_1$-collection", 2021) and Rathjen's analysis of $\Pi^1_2\mathrm{-CA+BI}$, I haven't noticed a treatment of iterated $\Pi^1_1$-reflection or non-Gandy ordinals. I'm far from an expert on ordinal analysis, so it's possible I've missed their appearance if they do appear. In the case they don't show up, how is analysis done without dealing with these phenomena, assuming they really do contribute so much to the structure of the admissible ordinals in that range?
