This paper revisits a classical, yet fundamental, discussion of theoretical computational linguistics: the computational complexity of natural languages. Past studies have revealed that syntax, as observed in Swiss-German, is not weakly context-free. Concerning morphology, Culy ( 1985 ) employed a construction in Bambara to show that morphology is not weakly context-free; however, Manaster-Ramer ( 1988 ) pointed out that the Bambara case can be problematic because the wordhood of the construction is reliant on special tonal behaviors, and it is ambiguous whether the behaviors belong to the morphological domain. This raises doubts about whether the case can be considered a genuine morphological phenomenon. In this paper, we argue that Classical Ainu, a language we examine, also defies weak context-freeness at the morphological level. The construction we introduce is unambiguously morphological because this language’s valency-sensitive structure and valency-changing operations, such as noun incorporation, preclude its grammatical interpretation as syntactic.