To be not so cheeky, I am giving a detailed answer for the following multiple version of   PR-theorem.

> **Theorem**: For any countable sequence $\{\mathbf{a}_n\}_n{\in \omega}$   non-zero Turing degrees, there is some degree $\mathbf{g}$ so that for any $n$, $\mathbf{a} _n\vee \mathbf{g}=\mathbf{g'}$.

I know four proofs of the theorem.

The first is by a slightly modification of Posner-Robinson's original argument. It was Joe who told me that $\Delta^0_2$ version of the theorem 
 can be found in the P-R's original paper. Then Frank Stephan observed that, by a slight modification of their method, one may obtain the (almost-)full version.

The second proof is about Jockusch-Shore style proof, which is due to Kirill Gura, who was an undergraduate student in Madison. I learned the proof from Joe too. In Kirill's proof, a notion called Kalimullin pair, which is from the  $e$-degree theory, was used.

The third proof is due to Stpehan, Tanuwijaya, Yang and me. It is also a Jockusch-Shore style proof. The key fact we used is that for any countably many non-zero degrees, each of them contains a set so that the union of them is immune but the intersection is infinite. 

The fourth-proof is due to Slaman. And it is a pretty straightforward application of Kumabe-Slaman forcing. The proof can also be found in Kirill's draft.

Notice that $\mathbf{g}$ can be 1-generic in the both the second and third proofs. And no way in the other proofs. Ted told me that he and Kucera tried very hard to find a method to replace Kumabe-Slaman forcing, which, so far, has very limited applications such as to prove higher P-R theorem.

I got interested in the theorem when I was in a Dagstuhl meeting in 2017. Joe told me that he was thinking about a question, which was raised by Andrew Marks,  about simultaneously  randomness relative to multiple-degrees. Then I just recognized that maybe a multiple version of PR-theorem useful. Then he told me that it is already in P-R's paper.  

That was all I remember.