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and to even know if such a thing is possible before pursuing it

I'm not the creator of that magnificent piece of programmatical art. But I can take a stab at this. As long as the languages are Turing complete it is possible.

Any Turing machine can simulate any other Turing machine so once you step up to that level the challenge is keeping your head through the maze of nuances between many of those languages. I think the logic behind it goes something like this, if a language is complete (eg in the Gödel sense) then you can have it produce things that are nonsense in its own language.

Anyways I would expect the bulk of the work to be making those incremental step of going from each language to the next. In terms of data shape, I am thinking a linked list. So the final step is to call them all in a nested fashion, bootstrapping each up onto the top of that stack as you go.

The quines are not simulating each other, they are just printing a string after some formatting (which usually involves embedding a copy of the string into itself). There's no Turing completeness needed (most of the quines don't even have branches), much less Gödel encodings.

they are just printing a string after some formatting

There's (at least) one language in quine-relay that's not that simple: Piet. It's a visual 2D language written as an image file.

https://www.dangermouse.net/esoteric/piet/samples.html

Oh huh you're absolutely right. Start at the bottom, work your way up

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