At the constructible hierarchy $L$, for each stage $L_\alpha$ we keep having subsets of it newly arising high up the hierarchy:
is it a theorem of $\sf ZFC + V=L$ that this stops by $L_{|\alpha|^+}$?
if yes, would that still hold for the case of relative constructivility $L(A)$ for any set $A$?
if neither, then is it consistent with $\sf ZFC+V=L(A)$ to have those stop by $L_{|\alpha|^+}(A)$?
The context for asking this question is related to an endeavour to prove Con(NF) via having some functions over a model of $L(A)$, and one of its requirement is at least the third one.
The details of the context is:
The idea is to work within a suitable extension of $\sf ZF-Reg.$ on a transitive non-well-founded model $M$ of $\sf ZF + V=L(A)$ where in it all stages $L_\alpha(A), L_\gamma(A)$ are of the same cardinality as long as $|\alpha|=|\gamma|$, but when $|\gamma| > |\alpha|$ we'll have $|L_\gamma(A)| > |L_\alpha(A)|$, and such that if $|\beta|$ is the successor of $|\alpha|$, then $|L_\beta(A)|=|\mathcal P(L_\alpha(A))|$ , now if it is consistent to have $\mathcal P(L_\alpha(A)) \subseteq L_\beta(A)$, then clearly we'll interpret $\sf NFU$ through the existence an external bijection $j: L_\beta (A) \to L_\alpha (A)$, where $\alpha, \beta$ are limits, and such that the relations $J,J^{-1}$ are set (i.e., $ \in M$) like, where generally for any function $f$ on $M$, the relation $F$ is defined as: $$ y \ F \ x \iff \exists z \in x: y=f[z]$$; where $f[z]=\{f(m) \mid m \in z\}$
The interpretation is through defining a new membership relation $\in^*$ over $L_\alpha(A)$ as:
$x \in^* y \iff x \in j^{-1}(y) \land j^{-1}(y) \subseteq L_\alpha(A)$
This would interpret $\sf NFU$, all elements of $L_\alpha(A)$ that are not images of subsets of $L_\alpha(A)$ under $j$ would be $\in^*$ empty, and so are the Ur-elements, but due to particulars of this construction, we have as many non-subsets of $L_\alpha(A)$ in $L_\beta(A)$ as subsets of $L_\alpha(A)$, and this would be copied internally inside $L_\alpha(A)$, and so we get $\sf NFU + |Set|=|Ur|$ which interprets $\sf NF$.
Of course this argument begs a proof of existence such external bijection $j$ in the first place, but even before that we need to capture the whole powerset of $L_\alpha(A)$ below some $L_\beta(A)$ where $|\beta|=|\alpha|^+$ in order to assure having as many sets as Ur-elements.