Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem) Gödel's constructible universe ($L$) is defined using definable power set operator in first order logic ($\mathcal{L}_{\omega ,\omega}$). One can produce such a universe in infinitary logics in the same way using corresponding notions of formulas and definability. Obviously $L$ becomes larger when the logic has more expression power.
For each cardinal $\kappa$ define $L_{\kappa}$ to be Gödel's constructible universe in the infinitary logic $\mathcal{L}_{\kappa,\kappa}$ and $L_{\infty}$ is Gödel's constructible universe in $\mathcal{L}_{\infty,\infty}$.
(1) Is $L_{\kappa}$ a model of $ZFC$ for each cardinal $\kappa$? What about $ZFC+GCH$?
(2) What is $L_{\infty}$?
(3) Is there a (possibly large) cardinal $\kappa$ such that $L_{\kappa}$ is Dodd-Jensen core model, $L[U]$, $HOD$, etc?
(4) What are the consistency strengths of the existence of non-trivial elementary embeddings from $\langle L_{\kappa},\in\rangle$ to itself for different $\kappa$s in the sense of infinitary logic $\mathcal{L}_{\kappa,\kappa}$? 
Note that by Prof. Hamkins' answer for $L_{\infty}$ finally it reach Kunen's inconsistency but what about a given cardinal $\kappa$? Are all these consistency strengths for different cardinals bounded by some large cardinal axiom and there is a gape between consistency strength of the existence of a non-trivial elementary embedding from $\langle L_{\infty},\in\rangle$ to itself and consistency strengths of the existence of such elementary embeddings for $L_{\kappa}$s?
(5) If there is a cardinal $\kappa$ such that $L_{\kappa}=HOD$, is it possible to determine consistency strength of the existence of a non-trivial (first-order) elementary embedding from $\langle HOD,\in\rangle$ to itself by analyzing the growth speed of the consistency strength of existence of such embeddings for $\langle L_{\lambda}, \in\rangle$s in $\lambda <\kappa$?
(6) What is $L_{\kappa}$ for the least strongly compact cardinal $\kappa$?
 A: Theorem. $L_\infty$ is the entire set-theoretic universe $V$.
Proof. I claim that every set will arise in the construction process, because eventually it will become explicitly definable by a formula. In infinitary logic, there are far more than only countably many formulas, and one can cook up a formula to define a specific set, by using the formulas that define its elements. What I claim specifically is that for every set $a$ there is a ${\cal L}_{\infty,\infty}$ formula $\psi_a(x)$,  such that in any transitive set $M$ with $a\subset M$ we have $a=\{ x\mid M\models\psi_a(x)\}$.  Suppose that this is true for each $a\in A$. Consider the formula  $$\psi_A(u)=\bigvee_{a\in A}(u=\{x\mid \psi_a(x)\}).$$
In any transitive $M$ with $A\subset M$, it follows that $\psi_A(u)$ will hold if and only if $u=a$ for some $a\in A$. Thus, $A=\{u\mid M\models \psi_A(u)\}$, and so $A$ is also definable. Thus, by $\in$-induction, we've verified that every set is definable, and so every set in $V$ eventually arises in your universe $L_\infty$. QED
This argument is similar to the fact that if one undertakes the constructible universe using second-order logic, rather than infinitary logic, the result is $\text{HOD}$.
Theorem.(Myhill & Scott) The constructible universe in second-order logic is the same as HOD. $$L_{SO}=\text{HOD}.$$
Proof. Any set that appears in $L_{SO}$ is ordinal definable, and so $L_{SO}\subset \text{HOD}$. Conversely, if $A$ is a set of ordinals in $\text{HOD}$, then it is ordinal definable in some $V_\alpha$, and so once $L_{SO}$ has constructed up to some stage $\theta$ of size at least $|V_\alpha|$, then in second order logic we can define $A$ as a subset by saying "there is a relation on $\theta$ which makes it isomorphic to $\langle V_\alpha,\in\rangle$, such that the formula is true for the corresponding ordinals. That is, the second-order logic allows us to summon a copy of $V_\alpha$ and run the definitions inside it. QED
For a reference, see Myhill, Scott, Ordinal definability. 1971 Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., (1967) pp. 271–278 Amer. Math. Soc., Providence, R.I. .
Your models $L_\kappa$ arise in Chang's paper in the very same journal issue:

C.C. Chang, Sets constructible using $L_{\kappa,\kappa}$, 1971 Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) pp. 1–8 Amer. Math. Soc., Providence, R.I.
MR Abstract. Let $C_\alpha^\omega$ be the αth level in the Gödel ramified hierarchy of constructible sets. The author studies the hierarchy $C_α^κ$ that results on using $L_{κκ}$ definability (in place of $L_{ωω}$ definability) in generating this hierarchy, where κ is a regular cardinal. The class $C_κ=⋃_αC_α^κ$ is a model of ZF; in fact, it is the smallest transitive model containing all the ordinals closed under <κ-termed sequences (Theorems I and II). It need not be a model of the axiom of choice (Theorem IV). Other results show what happens to the GCH and Scott's result on measurable cardinals in this model. These results depend on a generalization of Gödel's collapsing lemma (Theorem V). Finally, the author shows how to improve results about indiscernibles in $C_ω$ by using infinitary formulas. In particular, he shows that if there is a Ramsey cardinal then $C_{ω_1ω}$ is an $L_{ω_1ω}$-elementary substructure of $C_ω$, thus improving Silver's result.

The case $\kappa=\omega_1$ is known as the Chang model.
