In Chapter 1 of Oppenheim & Schafer's Digital Signal Processing, the authors state that not all discrete-time signals are the result of sampling continuous-time signals, and thus they are making the choice not to "force results from analog systems theory into a discrete framework" for the textbook.
Can anyone think of examples of naturally occurring digital signals that are not the result of sampling?
I believe that Oppenheim and Schafer were thinking of the creation (synthesis) of waveforms in making that statement: Simply, we need not sample an actual analog signal to create a digital signal. However the OP's question is interesting in limiting this further to "naturally occurring".
Without arguing too much about what "naturally occurring" actually means, consider these situations that could apply to being discrete in time (discrete-time signals), and finally to being both discrete in time and discrete in magnitude (digital signals):
Creating an animation by writing an image on each page of a book and flipping through the result, presents each image as a discrete sample.
Dripping water from a faucet creates sounds as discrete events.
Lightning is an example of a single impulse, and the image from a single lightning flash is a single sample.
Clapping, and finger snapping.
The above represent events at discrete events in time which are therefore "discrete-time signals". "Digital signals" have the property of being both discrete in time and discrete (quantized) in magnitude. Mortgage payments are an example of values that are discrete in time (paid on discrete days) and quantized in magnitude (to the closest penny).
Another example is taking a cross country trip and recording our distance to the closest mile on each day.
Please see this related post.
Look at yourself in the mirror! Gazillions of cells that constitute your body — though not all of them — have DNA strands, which can be viewed as (quaternary) digital signals of the type $$\{1,\dots,n\} \to \{ \texttt{A}, \texttt{C}, \texttt{G}, \texttt{T} \}$$ where the values are the $4$ nucleobases: adenine ($\texttt{A}$), cytosine ($\texttt{C}$), guanine ($\texttt{G}$), thymine ($\texttt{T}$).
Digital = discrete in values, discrete over domain (e.g. time, position; i.e. input space)
One can define a "more digital" signal that's not only discretely realized over a continuous domain, but whose domain itself is discrete - i.e. incapable of realizing most values. There, prior examples can be made qualifying by a redefinition of the domain - e.g. number of fish in a lake per year. Then, "the most digitalest" signal is one where a continuous definition, even if we wanted, is invalid:
