Hmm, I'd want three very different types of RPGs. I'd want to cover my main genre loves, and have at least one be a universal system. I'd also like it to be games I'd both want to run and play as I enjoy both parts.
Now, do I go by the title of desert isle or the post which is just a limit for the rest of time. The main difference here is the internet -- am I able to look things up, see others house rules, (use VTTs), get new supplements as they come out, etc.
For example, Fate is likely going to be my universal system. But really it's best supported in that role with the plentiful fan creations on the internet, just to give variety as opposed being stuck with how I (and the others on the island) would design.
I'm torn on PbtA - I really like bespoke games, and one of my genre loves is supers and my favorite supers game is currently Masks: A New Generation. However, it's super tight to one concept, and I think that is too tight for one of my three forever games. Generally, what I most love about PbtAs also makes them poor choices for a forever game, that they are very tightly tied to a concept.
Ooof, having to be forever games really cuts a bunch of my favorite RPGs out, because they are tightly mechanically tied to a concept that makes them superior for me, but not inferior when they are 1/3 of your total games you get to play. This really skews what I pick in an interesting way.
The question about supplements comes up for covering another genre love, heroic fantasy. My favorite currently is 13th Age, so I could take 13th Age 2nd ed with me. But D&D just has so much more published for it regularly that would give me more variety in a forever game. Those two exceed any of the other big tent fantasy RPGs I've tried, and no matter how good I've heard it to be I wouldn't risk a choice on a system I hadn't played or run.
I'm really torn on Supers. I played countless hours of Champions (now Hero System), and it's technically a good generic system as well. But I have no patience for such a heavy, crunchy game anymore. But here I have a list of systems I want to try, a bunch of systems I've tried and wouldn't make the list, and Masks: A New Generation which we already described the issue with. Funny, this is one where "on a desert island" is a benefit. If we have enough leisure time to play RPGs, maybe a crunchy and honestly a bit slow system will be fine. If I need to pick not just an RPG but an edition, I'd probably go with FRed (Fifth Revised) instead of 6th, though the BBB (Big Blue Book) was the one I spent the most time with and I might fall back to that. If I did I'd still leave Fate in there as a very different feeling generic system.