To be more specific than the short title, I try to gauge the size of some "normal" function spaces as e.g. found in functional analysis against set universe sizes at certain stages.
For the sake of the argument, let's think of an integral transformation such as $$ I(f)(x):=\int_0^x f(x+y) {\mathbb d}y $$ which we can read as a nicely behaved subset of $$(\mathbb R \to \mathbb R)\to(\mathbb R \to \mathbb R).$$ In terms of cardinality the latter is $$(|\mathbb R|^{|\mathbb R|})^{|\mathbb R|^{|\mathbb R|}}$$ and since $$|\mathbb R\to\{0,1\}|=|\{0,1\}|^{|\mathbb R|} = |{\mathcal P }\mathbb R | > |\mathbb R|$$ those space of transformation is already quite large.
Now on the other hand, let's look at the Neumann universe at a certain level. With $V_{\omega}$ we have the collection of all hereditarily finite sets and from he we want to apply the power set operation a finite number of times, e.g. $$V_{\omega+k}={\mathcal P }\,{\mathcal P }\,\cdot \cdot \cdot {\mathcal P }\,{\mathcal P }\,{\mathcal P }\,V_{\omega}$$ My issue is that the base of the exponential with ${\mathcal P }$ is "merely" what you get with exponentiation of $|\{0,1\}|$ but with function spaces I exponentiate (exponentials of) the reals.
The question is for what $\alpha$ a containing set $V_{\alpha}$ is "save", if I don't want to do induction of the exponential but stick to finite length expressions for functions spaces as the one above. I gather $V_{\omega+\omega}$ suffices for most math and the stuff I posted above, although it doesn't make for a closed universe when considering functions of bigger ordinals (out of scope of my question).
If I'm right with the above, desn't $V_{\omega+k}$ work too? Are there operations (or maybe, secondarily, proofs) that lead me out of a finite nuumer of powers of $V_{\omega}$?
Finaly, can does it suffice to consider constructive $_{\omega+k}$ or $_{\omega+\omega}$ instead of the $V$ variants at the same stage here? (I know that they coincide for just $L_\omega$.)
Generally speaking, that depends a lot on how you code things. You could encode the real numbers so that they appear, as the concrete object, at any arbitrarily chosen point in the hierarchy (e.g. you could say that the ordered pair $(a,b)$ is coded by $\{\{a\},\{a,b\},\alpha\}$ for some infinite ordinal $\alpha$, which ensures that anything whose definition goes through ordered pairs, like functions or equivalence relations, would have rank of at least $\alpha+1$).
If we ignore coding and only consider cardinality, i.e. "what would be the least rank in which we can encode such object", then the answer is that $V_{\omega+1}$ is the set of reals, and therefore the real numbers appear, as an object, in $V_{\omega+2}$.
Now, taking each power set brings you up by another level, as you pointed out. So the $\alpha$th power set of the real numbers is $V_{\omega+1+\alpha}$. If $\alpha\geq\omega$, this is just $\omega+\alpha$, and if $\alpha\geq\omega^2$ it's just $\alpha$.
If you just want the finite powers, then, $\alpha=\omega$, i.e. $V_{\omega+\omega}$, is your buddy, then. If you only care for a finite bound, $k$, then going for $k+1$ power sets, i.e. $V_{\omega+k+2}$, is enough.
Regarding the constructible hierarchy, though, this is a whole other thing. For one, $V=L$ is not a consequence of $\sf ZFC$, and for another $L_\alpha$ has the same cardinality as $\alpha$, whereas $V_\alpha$ can be enormously bigger than $\alpha$. Especially so in the cases you're interested in.