Rotations of regular polygons.
Start with an equilateral triangle. Label the vertices one, two, three. Define a rotation to be 60° counterclockwise, for example (or you could go clockwise). Note that the locations of the vertices have shifted. Consider this a new “state“ or “configuration”.
Now, what “configuration” does one end up with after n rotations?
Then, extend this to any n-sided polygon. A real life application would be modeling the movement of molecules. Gallian’s abstract algebra book shows a cross-section of a 28-sided HIV virus, for example. Or, how about rendering graphics in video games? Modeling movements of satellites and space shuttles in outer space?
A more accessible example would be a clock, or days of the week.
Incidentally, you could then use mod 12 or mod 7 to discuss notes on musical scales.