Example of Fourier Transform not existing for real-life signals?
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 

$E =\int_{-\infty}^{\infty}|x(t)|^2dt$ 

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.