You can take distributional derivatives of the devil's staircase. See Wikipedia for an explanation.

**Edit:** Sorry I was so curt. The answer to your revised question is no. The Wikipedia link above gives references to the fact that all distributions on the real line (and finite dimensional Euclidean spaces) are distributional derivatives of continuous functions, and continuous functions can be lifted canonically to measures.

As Rekalo mentioned, there is a notion of hyperfunction due to Sato that arises from studying boundary values of holomorphic functions, and hyperfunctions form a strictly larger space than distributions (in particular, essential singularities are allowed). I've heard of other spaces of generalized functions such as modules over sheaves of pseudodifferential operators, but I don't know how they relate to each other, or how one proves theorems with them.