Seems unlikely, given the existence of things like quines), and the fact that self-reference comes pretty easily. I recommend reading Godel Escher Bach, it discusses your original question in the context of this sort of self-referential mathematics, and is also very entertaining.
Quines don't say anything about human working memory limitations or the amount of time a human would require for learning to understand the whole system, and furthermore only talk about printing the source code not understanding it, so I'm not sure how they're relevant for this.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
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