Do any stream ciphers with aperiodic keystreams exist? Exactly what it says on the tin. I can imagine constructing such keystreams from:


The binary expansions of irrational numbers
Chaotic systems like the logistic map or the Lorenz attractor.
The bitmap representations of images.
Extracting the y-coordinate from equations such as $sin(x)$ or $e^x \mod 1$ every n increments.
Well, it's easy to show that any generated with a bound on the amount of internal state must eventually become periodic.  It's also easy, once we allow a generator whose internal state grows arbitrary large over time, to design a generator that never repeats (and the state growth required is actually quite reasonable).

That said, I don't know of any proposed stream ciphers that attempts to be completely aperiodic.  I suspect that it's mostly because aperiodicity is not a realistic requirement (as opposed to the nicety of being implementable within a bounded memory space).  After all, it is easy to have, even with a bounded state generator, such a large period (e.g. a period of $&gt;10^{100}$) that we never have to worry about it.  After all, if we never generate that much output, does it really matter if it would repeat if we did?