Let $n\in\omega$. Call a real $x$ $n$-special if $x\in L\_{\omega\_{n}^{CK,x}}$.
Are the $n$-special reals bounded in $L\_{\omega\_{1}}$? That is, is there $\beta<\omega\_{1}$
such that $L\_{\beta}$ contains all $n$-special reals?



A question about relativized admissibility theory
Let me assume first that by $\omega\_1$ you meant 
$\omega\_1^L$, that is, the $\omega\_1$ of $L$. In this case, no, even the $1$-special reals are not
bounded. To see this, note first that there are unboundedly many
$\gamma\lt\omega\_1^L$ with $\gamma$ countable in $L\_{\gamma+1}$.
(Any stage where a new real is added collapses everything to
$\omega$.) For such a $\gamma$, let $x$ be the $L$-least code of
$\gamma$, so that $x\in L\_{\gamma+1}\subset L\_{\omega\_1^x}$, and
so $x$ is $1$-special. Thus, in $L$ there are uncountably many
distinct $1$-special reals, and so they cannot be bounded.


But if you meant $\omega\_1$ to mean $\omega\_1^V$, then of course
it is consistent that $\omega\_1^L\lt\omega\_1^V$ and in this case
all the constructible reals, including all the $n$-special reals,
appear in $L\_\beta$ where $\beta=\omega\_1^L\lt\omega\_1$, which
would make them bounded in the sense you asked.


