This is not a high school question, but it seems to be too broad to fit in any of the other categories. Fractals are cute (nice pictures that can also be used to give better graphics, and also "shocking" that one can define a non-integer dimension), can be used to estimate lengths or volumes of irregular shapes (coastlines, Brownian motion or surfaces, fluid dynamics and all that), is a byword anytime anyone finds some self-similar scaling (everything from galaxies to quasi-crystals), and so forth. But do they lead to any other new mathematics? For example, the motion and logical paradoxes are also "cute", but they led to whole new and fruitful fields of mathematics being born. But, for example, are Hausdorff dimensions good for anything besides fractals? Is the study of fractals good only for itself?