Wolfram's big thing is 1D cellular automata. A simple class of automaton he seems to miss is 1D cellular automata in which the state of a cell depends not only on the state of cells above it, but on cells to the left of it. If a raster scan is used to generate the pattern, this doesn't violate causality.
Thinking in terms of hexagonal grid makes this seem a rather natural way to do things. A very basic class of such automata would have binary state and depend on the three anti-causal neighbours of a hexagonal cell. As with Wolfram's elementary cellular automata, there are 256 such automata, and they show a similar richness of behaviour.
If each row is limited to finite size, these can also be thought of in terms of 0D delay-line automata, which seems a particularly natural form.
Images showing results for all 256 such automata:
Such automata also have a more straightforward jigsaw-puzzle implementation than Wolfram's elementary cellular automata.
Ah, I tell a lie, he did play around with these. On page 1034. It really is quite a big book.