Let there be an omega sequence of ordinals such that the first is the least $\Sigma\_1$-admissible ordinal and the $n+1$st is the least $\Sigma\_{n+1}$-admissible ordinal. What is the name, if any, of the union of all of these? Incidentally, $L$ at the level of this ordinal would be the minimal model of ZFC minus the power set axiom.



Question on a limit of admissible ordinals
This ordinal is not as large as you claim. In particular, the constructible universe up to this ordinal does not satisfy ZFC minus power set, and the ordinal is not even admissible. 


To see this, let $\gamma\_n$ be the least $\Sigma\_n$-admissible ordinal (which I understand you to mean that $L\_{\gamma\_n}$ satisfies KP plus $\Sigma\_n$-replacement, or equivalently, $\Sigma\_n$-collection); and then the ordinal in question is $\gamma=\sup\_n\gamma\_n$ and the corresponding structure $L\_\gamma$. But this structure does not satisfy KP, since the map $n\mapsto\gamma\_n$ on domain $n\in \omega$ is $\Sigma\_1$-definable in $L\_\gamma$. The point is that whether some $L\_\beta$ satisfies a certain sentence is a $\Delta\_0$ espressible assertion about $L\_\beta$, since all quantifiers are bounded by $L\_\beta$, and whether $\beta=\gamma\_n$ is a $\Sigma\_1$-property of $\beta$ inside $L\_\gamma$. Namely, $\beta=\gamma\_n$ if and only if there is a transitive set $X$ that thinks it is $L\_\beta$ and there is a satisfaction predicate on this set fulfilling Tarski's recursive definition of truth and according to this satisfaction predicate, the set $X$ satisfies KP and every instance of $\Sigma\_n$-collection, and finally no smaller $\beta'\lt\beta$ satisfies all those. 


