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Not sure how this translates to "rules" in a cellular automaton, but: https://en.wikipedia.org/wiki/One-instruction_set_computer

This is exactly what I was asking about. Thank you!

There's also a few "one instruction" CPUs

You can build anything with AND and NOT gates or OR and NOT gates.

You can build anything with just NAND gates.

Well, we know that rule 110, which is like a 2d versions of Conway's game of life, is also Turing complete. So at least that simple.