The interesting thing about it is that it shows a very-bad-case for run-length encoding.

It's ironic that each item in the sequence is a sort of rle of the previous one. Granted, it is a very inept rle, since it causes the length of the "encoded" string to grow, but it is rle "in spirit". In fact, with a different starting point (e.g. 111111122222222), the rle would be initially OK (i.e. 7182). Of course, it would immediately start to grow again (17111812, 111731181112, etc.). It is also interesting to note that this "encoding" has one fixed point, namely 22, and it's not hard to convince oneself that it is unique. Anyway, I don't know much about rle, but I wonder if it is a property of every run-length encoding scheme that repeated applications of the scheme, starting with some string of integers, would lead to increasingly longer strings... But this is beginning to get very OT...