Interesting question: why does this (intuitively and irrationally) seem to me like a sadder fate than something like heat death?
The only long-term scenario I know that avoids both this outcome and heat death involves launching colonies into the "dust" as Greg Egan described. Unfortunately the assumptions required for that to work may well turn out to be false. There's no known law of nature saying we can't be trapped in the long term. And if we are trapped, I think I prefer a repeating paradise to heat death.
Conway’s Game of Life is Turing-complete. Therefore, it is possible to create an AI in it. If you created a 3^^3 by 3^^3 Life board, setting the initial state at random, presumably somewhere an AI would be created. Would this AI somehow take over the whole game board, if given enough time?
Would this be visible from the top, as it were?
EDIT: I probably meant 3^^^3, sorry. Also, by generating at random, I meant 50% chance on. But any other chance would work too, I suspect.