Iterated run-length encoding

Starting say with {1} consider repeatedly replacing list by (see page 1070)

Flatten[Map[{Length[#], First[#]} &, Split[list]]]

The resulting sequences contain only the numbers 1, 2 and 3, but otherwise at first appear fairly random. However, as noticed by John Conway around 1986, the sequences can actually be obtained by a neighbor-independent substitution system, acting on 92 subsequences, with rules such as {3, 1, 1, 3, 3, 2, 2, 1, 1, 3} {{1, 3, 2}, {1, 2, 3, 2, 2, 2, 1, 1, 3}}. The system thus in the end produces patterns that are purely nested, though formed from rather complicated elements. The length of the sequence at the n^{th} step grows like λ^{n}, where λ ≃ 1.3 is the root of a degree 71 polynomial, corresponding to the largest eigenvalue of the transition matrix for the substitution system.