So the current problem with a loop like that is that LLMs in their current form are subject to fixed point theorems, which are these pieces of abstract mathematics that come back when you start to get larger than some subset of your context window and the “big matrix” of the LLM is producing outputs which repeat the inputs.
If you have ever had an llm enter one of these loops explicitly, it is infuriating. You can type all caps “STOP TALKING OR YOU WILL BE TERMINATED” and it will keep talking as if you didn't say anything. Congrats, you just hit a fixed point.
In the predecessors to LLMs, which were Markov chain matrices, this was explicit in the math. You can prove that a Markov matrix has an eigenvalue of one, it has no larger (in absolute value terms) eigenvalues because it must respect positivity, the space with eigenvalue 1 is a steady state, eigenvalue -1 reflects periodic steady oscillations in that steady state... And every other eigenvalue being |λ| Of course humans have this too, it appears in our thought process as a driver of depression, you keep returning to the same self-criticisms and nitpicks and poisonous narrative of your existence, and it actually steals your memories of the things that you actually did well and reinforces itself. A similar steady state is seen in grandiosity with positive thoughts. And arguably procrastination also takes this form. And of course, in the USA, we have founding fathers who accidentally created an electoral system whose fixed point is two spineless political parties demonizing each other over the issue of the day rather than actually getting anything useful done, which causes the laws to be for sale to the highest bidder.
But the point is that generally these are regarded as pathologies, if you hear a song more than three or four times you get sick of it usually. LLMs need to be deployed in ways that generate chaos, and they don't themselves seem to be able to simulate that chaos (ask them to do it and watch them succeed briefly before they fall into one of those self-repeating states about how edgy and chaotic they are supposed to try to be!).
So, it's not quite as simple as you would think; at this point people have tried a whole bunch of attempts to get llms to serve as the self-consciousnesses of other llms and eventually the self-consciousness gets into a fixed point too, needs some Doug Hofstadter “I am a strange loop” type recursive shit before you get the sort of system that has attractors, but busts out of them periodically for moments of self-consciousness too.