A rather crazy (and very useful) example is a fundamental solution of an arbitrary differential equation with constant coefficients, i.e., a distribution $u$ satisfying $P(D)u=\delta_0$ where $P$ is a polynomial and $D$ is the differentiation operator. The construction can be found in many decent PDE textbooks. It is as far from the standard "take a non-smooth function, differentiate a few times" idea of how to get distributions as possible.

Another thing to understand is that, like with everything else, it is even more important to learn what you can and what you cannot do with distributions than what they can be.