Is there an example of an infinite signal?
After the proposed heat death of the universe, no active life form will be available to measure the rest of any given signal. That's not a real world to anyone.

One could define a signal not as function of time but as function of location, and wrap around the location. An example is the present ambient temperature as function of distance traveled to the east from Pontianak, Indonesia, on the Equator. We don't know if the universe is infinitely large so measurements on a straight line would be a bit iffy.