ASR-complete is, by analogy to NP-completeness in complexity theory, a term to indicate that the difficulty of a computational problem is equivalent to solving the central Automatic Speech Recognition problem, i.e. recognize and understanding spoken language. Note that unlike NP-completeness, this term is typically used more informally.
Such problems are hypothesised to include:
These problems are easy for humans to do (in fact, they are described directly in terms of imitating humans). Some systems can solve very simple restricted versions of these problems, but none can solve them in their full generality.