Simplest discrete interactable universe Consider the set of all mathematically possible universes.
Take only the deterministic ones, that run forward in a single dimension of time, the state at a given instant being uniquely determined by the previous instant.
Take only the discrete ones, that don't require real numbers, e.g. omitting Newtonian physics. (For any continuous universe, there are discrete counterparts that approximate  it to finite precision, e.g. Newtonian physics simulated with 128-bit floating point. Though harder to formalize, let us say the intent here is to consider only universes that are naturally discrete, omitting those that are clearly trying to approximate real numbers.)
Lots of things are still candidates: cellular automata, Turing machines etc.
The next qualification is even harder to formalize, but I think it is intuitively reasonable: take only physics-like universes, in the sense of ones that don't have any notion of processors that can read memory, as primitives; where such things, if they are to exist, must be built from simple atomic entities that interact locally. So we still have cellular automata, but not Turing machines.
Now take only the ones that are Turing complete. Some cellular automata meet this criterion, e.g. Conway's Game of Life.
The final qualification is the trickiest one: consider those universes that are interactable, that permit the existence of embedded agents that can effectively sense and modify their environment.
The only deterministic, physics-like universe that I know of that is interactable (where 'deterministic' excludes e.g. Minecraft and 'physics-like' excludes e.g. Core Wars) is the universe we inhabit (which is excluded from consideration in this discussion by not being discrete).
I was prompted to post this question by stumbling on a discussion of attempts to find an existence proof that Conway's Game of Life is interactable: https://www.conwaylife.com/forums/viewtopic.php?p=137171&sid=7a1ea26cb487642c07ec42c51fa89e17#p137171
In particular, looking carefully at the above thread, it is clear that no one has found a way to do this. I conjecture that Conway's Life is not interactable. (To prime intuition on the matter, imagine living in a universe where the only way to sense your environment was to fire a beam of anti-protons in some direction. If nothing happens, there was nothing in that direction; if you find yourself dying of radiation poisoning, there was something, but you will never know what it was.) It's formally an open question; at the least, despite some decades of trying, we have not found a proof that it is.
Is there any known mathematical universe with the above properties, that is interactable? If so, what is the simplest one that is known? If not, has any progress being made on finding a proof of the existence or nonexistence of such?
 A: I think, if you want a mathematical answer, you will probably need to formalize some of your questions a bit more.
There’s been a lot of research in computer science about the computing power of various models, but it is unclear to me whether, for example, a pushdown automaton would be “physics-like” enough for you.
Similarly, I’m not certain what you mean by “interaction.”  The example you give seems to be getting at some property I don’t think it’s settled that human beings have.  The surface of the Earth is completely bounded by environments that will kill a human being.  Or could I go explore the country to my south, come to understand it, and not become a different person?  It sounds to me like you are thinking less of something like that, and more that some part or pattern of the universe be able to change it and not collapse into some high-entropy state?
It doesn’t seem like the fundamental particles of our universe have that kind of persistence.  They’re created and destroyed all the time.  We can’t point to any two particles at different points in time and say, “These are the same particle, distinct from all other particles.” And would a Buddhist agree that humans remain the same person leter, either?  If we’re actually patterns of completely-interchangeable, transitory protons, neutrons and electrons, who are we to say patterns in the Game of Life are not concrete enough for us?
This is more a philosophical reply than a mathematical one, I’m afraid.  Mainly because I’m not sure how to formulate the question as mathematical statements.
