I don't see a problem with this at all. Some processes -- e.g., Newton's iteration of the square root -- pushes _all of the information_ (except for the sign) into lower and lower order bits. You can recover these bits by running the Newton process backwards. This backwards Newton process will reach further & further into the "innards" of the initial number -- i.e., pulling out lower & lower order bits. The only way you can get this process to cycle is if you started with a rational number (which has a repeating expansion). A simple "bounded" system which is composed of a finite set of real numbers, will have an unbounded number of bits to fool with. I can envision such a system which recycles larger & larger numbers of bits from the initial condition real numbers, but stays bounded because all the real numbers are a priori bounded by some fixed constant. For example, take some real numbers whose integer parts are all equal to 3. These real numbers remain bounded in the range (3,4), but we simply cycle larger & larger numbers of the fractional bits. This system is chaotic, non-periodic and bounded. At 05:04 PM 3/15/2013, Bill Gosper wrote:

The idea that such a simple, bounded system can produce endless novelty is a little hard to swallow.