It's well known that Sacks forcing constructs a real of minimal constructability degree, i.e. a real $x$ such that for any $y\in L(x) \setminus L$, $L(y) = L(x)$. It's also well known that certain objects, such as $0^\sharp$, can never be created by a forcing extension.


Given these two facts there is a natural question:



> 
> Can there exist a real $x\notin L$ such that any inner model $W$ with $L(x) \supseteq W \supseteq L$ either $W=L$ or $W= L(x)$ but such that $L(x)$ is not a (set) forcing extension of $L$?
> 
> 
> 


Obviously we could also ask the question regarding larger sets than reals or class forcing extensions.



Is there a minimal extension of $L$ that is not a forcing extension?
Yes, this is possible and follows from Sy Friedman's paper [Minimal coding](https://www.sciencedirect.com/science/article/pii/016800728990002X?via%3Dihub) (you may better look at [Fine structure and class forcing](https://www.degruyter.com/view/product/151373)).


It follows from the results of the above
 paper that there is an $L$-definable class forcing for producing a real $R$ which is minimal over 
$L$ but is not set-generic over $L$.


