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.