As single waveforms go, you could construct one of these:
http://en.wikipedia.org/wiki/Weierstrass_functionThe series is not harmonic in the usual way, having energy at evenly spaced points -- the frequencies are exponential in n -- but the frequencies are still harmonics in the definitive sense (exact integer multiples of the fundamental). If a = b, the harmonics drop off as 1/f, comparable to a square wave (give or take the phase shift, and with a whole bunch of harmonics cut out).
It wouldn't be very exciting, though: the exponential dependency of both amplitude and frequency on n means you only get a few frequency components before you run out of gain-bandwidth in a real system.
You can also construct semi-fractal things, like Perlin noise, which is handy for images (an alpha-blended grayscale mask looks conspicuously like clouds, or a variety of natural textures), and should have a similar 'feel' aurally (i.e., as sound) or temporally (e.g., as intensity of light -- perhaps a peculiar flicker generator?).
As for fractal methods, generally speaking, because analog signals are continuous value and continuous time, you must employ some other mechanism to achieve the iteration effect. It can be sampling (as in the computed Mandelbrot example), event triggered (which is effectively equivalent; observe Logistic Map behavior in the peak-current-mode flyback controller), or cascaded (each function stage representing an 'unrolling' of a recursive function call).
On an even more fundamental level, you're talking Lambda Calculus: a system where integers have no place (sound familiar?), and iteration and counting is generally implemented recursively. (Sadly, we have to build each and every function we recurse over; a reused function must be re-implemented as well...)
An example of the latter construction: RF log detector. Just a chain of amplifiers of fixed gain and simple saturation behavior. As each one saturates, it ceases amplifying the signal, so that the final output becomes strongly limited, even for very weak inputs. The number of stages in saturation corresponds to the log of the signal strength; typically a diode detector reads each stage's amplitude, and a summing amp produces the output calculation. The transfer function is a little lumpy (because it's linearly interpolated between any two amplifiers nearing saturation), but in practice, not too bad (within a few dB). Rather than recursing forever, the number of stages is conveniently limited by available or desired gain-bandwidth and the noise floor.
Tim