This sequence has been well studied by John Conway. Years ago I read his article in Eureka, the magazine of the mathematical society of Cambridge University, but I don't know if you can easily get hold of it now.
You can find out a bit more here. In particular Conway shows that in the limit the sequence grows in length, on average, by a factor of $\lambda = 1.3035...$ at each step, with $\lambda$ a solution to a degree 71 polynomial. He also showed that there are essentially just 92 subsequences that repeat over and over again and never interact with each other. He (naturally) named these after chemical elements.
