I got curious based on this question here, but basically, is there ever a real-life signal that exists where its Fourier transform does not exist? If a signal is not finite energy, then its Fourier Transform does not exist, so what might be an example, (if any), of such a signal in real-life?
All real-life signals are finite energy. The universe contains a fixed (and finite) quantity of energy, which has been unchanged since it came into being.
A signal's energy is given by
Thus, the only way to make a signal's energy go to infinity is to allow it to continue for infinite time or reach an infinite peak level. While useful in mathematical and/or physics theory, neither of these is possible in reality.