I remember being blown away when I first read Mandelbrot's "The Fractal Geometry of Nature" about how utterly reasonable the definition of fractal dimension is. Moreover, as the title of the video suggests, I think that was the first time I saw that the original motivation for defining fractals did not have self-similarity as the central focus, but instead centered around capturing the idea of roughness that persists at all levels.
If I could revisit the video, one thing I might change is how I discuss the definition of a fractal. I wanted something concrete that a viewer could come away with, a way to fill in the blank as they say "fractals are not self-similar, instead they are _______". But I fear that focusing on non-integer dimension might not do appropriate justice to the nuances in trying to define roughness. I opted not to say that fractals are shapes whose fractal dimension exceeds their topological dimension because I didn't want to get into topological dimension. But even that definition misses a few rough objects. A rather nice analogy that one commenter referenced is that defining fractals for mathematicians is a bit like defining life for biologists: We all know what it is that we're trying to encapsulate, but every time you try to nail it down with a concrete definition it seems to neglect some of what you want or to admit some of what you don't want.