Over this past weekend I listened to an episode of This American Life titled Pro Se. Although the episode is nominally about people defending themselves in court, the first act of the episode was about a man who pretended to act insane in order to get out of a prison sentence for an assault charge. There doesn't appear to be a transcript, so I'll summarize here first.
A man, we'll call him John, was arrested in the late 1990s for assaulting a homeless man. Given that there was plenty of evidence to prove him guilty, he was looking for a way to avoid the likely jail sentence of five to seven years. The other prisoners he was being held with suggested that he plead insanity: he'd be put up at a hospital for several months with hot food and TV and released once they considered him "rehabilitated". So he took bits and pieces about how insane people are supposed to act from movies he had seen and used them to form a case for his own insanity. The court believed him, but rather than sending him to a cushy hospital, they sent him to a maximum security asylum for the criminally insane.
Within a day of arriving, John realized the mistake he had made and sought to find a way out. He tries a variety of techniques: engaging in therapy, not engaging in therapy, dressing like a sane person, acting like a sane person, acting like an incurably insane person, but none of it works. Over a decade later he is still being held.
As the story unravels, we learn that although John makes a convincing case that he faked his way in and is being held unjustly, the psychiatrists at the asylum know that he faked his way in and continue to hold him anyway, though John is not aware of this. The reason: through his long years of documented behavior John has made it clear to the psychiatrists that he is a psychopath/sociopath and is not safe to return to society without therapy. John is aware that this is his diagnosis, but continues to believe himself sane.
Similar to trying to determine if you are anosognosic, how do you determine if you are insane? Some kinds of insanity can be self diagnosed, but in John's case he has lots of evidence (he has access to read all of his own medical records) that he is insane, yet continues to believe himself not to be. To me this seems a level trickier than anosognosis, since there's no physical tests you can make, but perhaps it's only a level of difference significant to people but not to an AI.
Edited to add a footnote: By "sane" I simply mean normative human reasoning: the way you expect, all else being equal, a human to think about things. While the discussion in the comments about how to define sanity might be of some interest, it really gets away from the point of the post unless you want to argue that "sanity" is creating a question here that is best solved by dissolving the question (as at least one commenter does).