[marginalia_nu]

You see it in processes where something spreads to its vicinity.

This isn't really a natural example, but if you draw a square on a sheet of graphing paper. Next iteration you fill in each adjacent square. Repeat this process until you get tired of it. The radius increases linearly at a constant rate, but the area, the number of squares, as a function of each iteration, is growing quadratically.

Take a circular forest in a place where there are no fires and no logging. Its rate of growth is proportional to its circumference, which is proportional to its radius. Its area as a function of time is a quadratic function.