We propose the combination of two known extensions of Dung’s abstract Argumentation Frameworks (AFs): Incomplete Argumentation Frameworks (IAFs), where the existence of arguments and attacks may be uncertain, and Higher-Order Argumentation Frameworks (HOAFs), where attacks can be directed not only towards arguments but also towards attacks. As a result, we obtain Incomplete Higher-Order Argumentation Frameworks (IHOAFs), where attacks on attacks can also be uncertain. We take some important initial steps in the formal understanding of this novel framework by providing: (i) the definition of a conservative notion of completion, the hypothetical removal of uncertainty used to reason about argument acceptability in other formalisms for incomplete argumentation; (ii) complexity results for argument acceptability problems; (iii) an analysis of the previous two points in a well-behaved variant of dynamic logic.