Okay, so let's define a random function $F$, such that the value of $F(x)$ is uniformly distributed on $[-1,1]$, and such that for any $x$ and $y$ with $x \ne y$, $F(x)$ and $F(y)$ are independent. Now define $f(x)=\int_0^xF(x)\mathrm d x$.
If I remember right, $f$ will have highly pathological properties with probability one. Being a fractal is one of its tamer properties.
I also remember it had a specific name (maybe not this exact function, but based on it.) What was it?
Note: I may have gotten the definition wrong. Maybe what is was is you take as a random variable all the functions which are continuous with derivatives between $-1$ and $1$.