Few days I asked this question (https://math.stackexchange.com/questions/2907733/simple-ordinal-question) on MSE. Summary of the question is that I defined a certain function over ordinals $x \mapsto \beta_x$ (where $x<\omega_1$) and asked about its relation to the function $x \mapsto \omega^{CK}_x$. And it seems they are "nearly" the same (I mean the values, not the definitions) with the former function being continuous while the latter function is discontinuous (it does obey continuity condition on some limit values). In particular, $\omega^{CK}_1=\beta_0$ and $\omega^{CK}_2=\beta_1$.

Since asking that question, another question has come to my mind. Before getting into detail I will just briefly state the statement of question: **"Can we rigorously define an ordinal $\gamma$ which (intuitively) is to $\omega_1$ what $\omega^{CK}_2$ is to $\omega^{CK}_1$?"**

Here is the main motivation for the question. The (fictional) idea is that suppose completely "fictitiously" that we indeed have some well-order (of $\mathbb{N}$) with order-type $\omega_1$. Now suppose someone asked me whether this given $\gamma$ (which we are trying to describe rigorously) is greater than say $\omega{_1}^{\omega_1}$? My answer would be yes. And my "reasoning" would be that if somehow one just had access to the fictitious well-order just described, then using that hypothetical orcale function there also exists an ordinary program that describes the well-order (of $\mathbb{N}$) with order-type $\omega{_1}^{\omega_1}$. By the same reasoning $\gamma$ has to be greater than, say, the first fixed point of $x \mapsto (\omega_1)^x$. And so on...

I hope the drift of the question is clear at this point. The question is that what would be a reasonable rigorous definition of $\gamma$ that also corresponds well with this intuition.

**Edit:** Based upon suggestion in comments, I posted the side question separately: Relation of $\omega_{\omega_1+1}^{CK}$ to some other ordinals