"Largish" cardinals In what follows, $\mathsf{ZCKP}$ refers to the subset of $\mathsf{ZFC}$ consisting of the axioms of Zermelo set theory with choice and foundation ($\mathsf{ZC}$) plus those of Kripke-Platek set theory ($\mathsf{KP}$): equivalently, as these two theories have much overlap, $\mathsf{ZCKP}$ consists of $\mathsf{ZC}$ plus the axiom of replacement for $\Sigma_1$-formulæ (or equivalently, $\mathsf{KP}$ plus choice, foundation, powerset, infinity, and full separation).  A "largish" cardinal property means, informally, one whose existence is provable in $\mathsf{ZFC}$ but not in $\mathsf{ZCKP}$.
Here is the simplest example of a largish cardinal notion.  A theorem of Azriel Lévy (see, e.g., Barwise, Admissible Sets and Structures (1975), theorem II.3.5 on page 53 and theorem II.9.1 on page 76) states that for every uncountable cardinal $\kappa$, if $H(\kappa)$ is the set of sets hereditarily of cardinality $<\kappa$, then $H(\kappa)$ is a $1$-elementary submodel of the universe (meaning that every $\Sigma_1$ formula with parameters in $H(\kappa)$ is true [in $V$] iff it is true in $H(\kappa)$).  This implies that $H(\kappa)$ satisfies $\Delta_0$-collection (eqvt. $\Sigma_1$-replacement), and if $\kappa$ is a strong limit, then $H(\kappa) \models \mathsf{ZCKP}$ (and the converse is clear).  In particular, $\mathsf{ZCKP}$ does not prove the existence of strong limit cardinals.
Now I am interested in strengthenings of this condition on $\kappa$ such that the existence of these cardinals is still provable in $\mathsf{ZFC}$.  Two obvious candidates are:


*

*$H(\kappa)$ satisfies $\mathsf{ZC}$ plus replacement for $\Sigma_n$ formulæ,

*$H(\kappa)$ is an $n$-elementary submodel of the universe (i.e., every $\Sigma_n$ formula with parameters in $H(\kappa)$ is true iff it is true in $H(\kappa)$).
These should at least imply that $\kappa$ is a fixed point of the beth function, so that in fact $H(\kappa) = V_\kappa$.  (Perhaps this should be added as a precondition to be worthy of the term "largish cardinal".)
Edit (on 2015-12-11, following the answer by Joel David Hamkins): I didn't realize how very different the two notions above are: the first (call them "$\Sigma_n$-replacing" cardinals) is "local" in that it involves only sets from $V_\kappa$ and can thus be expressed as a $\Delta_1$ property of $V_\kappa$, whereas the latter ("$\Sigma_n$-correct cardinals") is "global" and involves the entire universe.  This has a consequence of size: the smallest $\Sigma_2$-correct cardinal, as explained in Joel's response, is larger than the first $\Sigma_n$-replacing cardinal for all $n$, or even the first inaccessible, etc., and there is no hope of "computing" it.  There may be some hope for the first $\Sigma_2$-replacing cardinal, however.  I really should have asked two different questions.
My question is this: Can these conditions, at least for $n=2$, or perhaps some related ones, be rephrased in purely cardinal-theoretic terms (without appealing to model theory and if possible avoiding the Lévy hierarchy)?  Even better, can the smallest cardinal satisfying such a condition be "described" or "computed" in some way?  (In the same way that $\beth_\omega$, or "the limit of the sequence defined by $\kappa_0 = \omega$ and $\kappa_{n+1} = \beth_{\kappa_n}$" are descriptions/computations of the smallest strong limit cardinal and the smallest fixed point of the beth function.)
More generally, any comments on these or related properties would be welcome (including a better term than "largish cardinal").  There is probably some connection with powerset-admissible ordinals, although the exact relation escapes me.
One reason why one might be interested in such cardinals is that the corresponding $H(\kappa)$ might serve as a drop-in replacement for Grothendieck universes in a $\mathsf{ZFC}$ formulation of category theory (they are not fully Grothendieck universes, but the point is that the use of a construction that escapes from such a "universish" set is likely to be so rare as to be very conspicuous; and unlike Grothendieck universes, their existence follows from $\mathsf{ZFC}$).
 A: Your cardinals are known as the $\Sigma_n$-correct cardinals, and they arise in diverse set-theoretic contexts. For example, we use them extensively in our paper:


*

*J. Bagaria, J. D. Hamkins, K. Tsaprounis, T. Usuba, Superstrong
and other large cardinals are never Laver indestructible, to
appear in the Archive for Math Logic.


It is a ZFC theorem that the $\Sigma_n$-correct cardinals form a
closed unbounded proper class often denoted $C^{(n)}$.
One subtle point about the $\Sigma_n$-correct cardinals is that
although we have a concept of $\Sigma_2$-correct and
$\Sigma_3$-correct and so on, $\Sigma_n$-correct for any
particular $n$, there is no uniform-in-$n$ way to express the
concept of $\Sigma_n$-correctness in first-order set theory. The
concept is uniformly expressible in some second-order set
theories, such as Kelley-Morse set theory, which prove that there
is a truth-predicate for first-order truth.
Concerning your question, when $n\geq 2$ there can be no way to
define what it means for $\kappa$ to be $\Sigma_n$-correct by
looking only below $\kappa$, say, as a limit process, since such a
property would be too simple, as it could be verified inside
$V_\kappa$ itself, but such verifiable-in-$V_\kappa$ properties have complexity at worst $\Delta_2$. For example, the property of being $\Sigma_n$-correct cannot be
$\Sigma_n$-expressible, for then the assertion "There is a
$\Sigma_n$-correct cardinal" would reflect from $V$ to $V_\kappa$,
even when $\kappa$ is the least $\Sigma_n$-correct cardinal, which
gives a contradiction since there are none below the least one. Meanwhile, the property of being $\Sigma_n$-correct is $\Pi_n$-expressible, since one need only say that all the instances of $\Pi_n$ truth in $V_\kappa$ are actually true.
The case of $\Sigma_2$-correct cardinals is particularly attractive, and perhaps this is an example that interests you. The $\Delta_2$ properties are precisely the properties that are local, in the sense that they can be determined in any sufficiently large $H(\theta)$. You can read more on my blog post: 


*

*Local properties in set theory
It follows that a cardinal $\kappa$ is $\Sigma_2$-correct, if whenever there is an object having a certain properties inside some possibly very large $H(\theta)$, then there is such an object inside such an $H(\theta)$ with $\theta<\kappa$. In other words, $\kappa$ is $\Sigma_2$-correct, if whenever anything verifiable happens anywhere, then it happens inside $V_\kappa$. Alternatively, everything verifiable has already happened by the time you get to $H_\kappa$. Such a way of understanding $\Sigma_2$-correctness is extremely useful, since it aligns with how set theorists often think about verifying set-theoretic facts. 
(A small matter: the distinction between $V_\kappa$ and $H_\kappa$ disappears once $n\geq 2$, since in this case the cardinals are $\beth$-fixed points and so $V_\kappa=H_\kappa$.)
Lastly, your idea of using the $\Sigma_n$-correct cardinals as a
universe replacement idea is well known. This is known as the
Feferman theory, and I also discussed it here on
MathOverflow in my answer to the question What interesting/nontrivial results in Algebraic geometry require the existence of universes?.
