What annoys me about linguistics is that higher level languages are quite poor for basic human communication. Non verbal communication is maybe the most important channel.
Furthermore, all known human languages allow you to attribute utterances to another, earlier speaker, what is known as reported speech (often, though not always as Pirahâ demonstrate, through an embedded clause "He said that X").
Human non-verbal communication (which is EMPHATICALLY NOT to be confused with sign languages), on the other hand, is typically not very expressive. Apart from the admittedly very important, but nevertheless quite specialized domain of human emotion (conveyed by facial expression for instance), it is more or less inadequate for much of what we want to communicate.
I really agree about the abstract possibilities of speech, but I a bit annoyed at the fact in one's existence, the non verbal cues are so important, yet so often boldly ignored, and almost never taught.
It makes me kind of sad when people assume that "Chomsky's work = the whole of linguistics".