The following results of Mack Stanley might be of some interest:

Theorem 1. Let $L$ be a minimal countable standard transitive model of $ZFC$. There exists a real $x$ having the following three properties:

(1) $x\notin L_\alpha$.

(2) $L_\alpha[x]\models  ZFC$.

(3) $x$ is not definably generic over any outer model of $L$ that does not already contain $x$.

The proof of the above theorem is given in ``A non-generic real incompatible with $0^♯$. Ann. Pure Appl. Logic 85 (1997), no. 2, 157–192.''

The next theorem gives a result complementary to Theorem 1, namely, that every outer model of a sufficiently non-minimal universe is a generic extension of that universe with respect to the language of set theory.

Theorem 2. Suppose that $W$ is a countable standard transitive outer model of $V$, and that there exists a branch $B$ through $U$ such that $sup(B) = \infty = W \cap OR$ and $(W; V,B)$ satisfies $ZFC.$ Then there exists a $(V ;B)$-definable partial ordering $P$ and a filter $G$ on $P$ such that $G$ is generic over $(V ; P)$ and $W = V [G].$

Here $U=\{ u(\kappa): \kappa$ is a cardinal $ \},$ $u(\kappa)=\{ \lambda\leq \kappa: \lambda$ is a cardinal and$ H_\lambda=Skolem Hull_{Hyp(H_\kappa)}(H_\lambda) \cap H_\kappa \}$, and for any set $X,$ $Hyp(X)$ is the smallest admissible set with $X$ as an element.

The proof is given in ``Outer models and genericity. J. Symbolic Logic 68 (2003), no. 2, 389–418. ''

Note that for example, it follows that any countable model $W$ of $ZFC +$ $0^\sharp$ exists” is a generic extension of $L^W$. More precisely:

Corollary. Suppose that $W$ is a countable standard model of $ZFC +$ $0^\sharp$ exists”. Then there exists a $W$-definable, $L$-amenable partial ordering $P$ and a filter $H$ such that $H$ is generic over $(L; P)$ and $V = L[H].$

I may mention that the notion of genericity in the above results is different from the ordinary definition of genericity. See remark 1.3 of the second mentioned paper.

The following results of Mack Stanley might be of some interest:

Theorem 1. Let $L$ be a minimal countable standard transitive model of $ZFC$. There exists a real $x$ having the following three properties:

(1) $x\notin L_\alpha$.

(2) $L_\alpha[x]\models ZFC$.

(3) $x$ is not definably generic over any outer model of $L_\alpha$ that does not already contain $x$.

The proof of the above theorem is given in ``A non-generic real incompatible with $0^♯$. Ann. Pure Appl. Logic 85 (1997), no. 2, 157–192.''

The next theorem gives a result complementary to Theorem 1, namely, that every outer model of a sufficiently non-minimal universe is a generic extension of that universe with respect to the language of set theory.

Theorem 2. Suppose that $W$ is a countable standard transitive outer model of $V$, and that there exists a branch $B$ through $U$ such that $sup(B) = \infty = W \cap OR$ and $(W; V,B)$ satisfies $ZFC.$ Then there exists a $(V ;B)$-definable partial ordering $P$ and a filter $G$ on $P$ such that $G$ is generic over $(V ; P)$ and $W = V [G].$

Here $U=\{ u(\kappa): \kappa$ is a cardinal $ \},$ $u(\kappa)=\{ \lambda\leq \kappa: \lambda$ is a cardinal and$ H_\lambda=Skolem Hull_{Hyp(H_\kappa)}(H_\lambda) \cap H_\kappa \}$, and for any set $X,$ $Hyp(X)$ is the smallest admissible set with $X$ as an element.

The proof is given in ``Outer models and genericity. J. Symbolic Logic 68 (2003), no. 2, 389–418. ''

Note that for example, it follows that any countable model $W$ of $ZFC +$ $0^\sharp$ exists” is a generic extension of $L^W$. More precisely:

Corollary. Suppose that $W$ is a countable standard model of $ZFC +$ $0^\sharp$ exists”. Then there exists a $W$-definable, $L$-amenable partial ordering $P$ and a filter $H$ such that $H$ is generic over $(L; P)$ and $W = L[H].$

I may mention that the notion of genericity in the above results is different from the ordinary definition of genericity. See remark 1.3 of the second mentioned paper.