I'm wondering if there are any known result for the maximum large cardinal strength 
which can be preserved by Radin forcing? For instance, with any large cardinal hypothesis in the ground model, can one show that in $V[G]$ which is a generic for a Radin forcing of some length $\kappa$ is $V\_{\kappa + 2}$-strong?



Radin forcing preserving large cardinals
If you start with a strong (or a supercompact cardinal) and if you force with Radin forcing
$\mathbb{R}\_u$, for some suitable $u$, then you can preserve the full strength (or supercompactness) of $\kappa$.


To preserve partial strength, (weak) repeat points are sufficient (see Radin's paper [Adding closed cofinal sequences to large cardinals](https://www.sciencedirect.com/science/article/pii/0003484382900237?via%3Dihub)).


To get the full strength, see for example the proof of Theorem 2.14 of my paper with Gitik [Adding a lot of Cohen reals by adding a few I](http://math.ipm.ac.ir/~golshani/Papers/Addin%20a%20lot%20of%20Cohen%20reals%20by%20adding%20a%20few%20I.pdf).


