Any mental illness may be difficult to treat for any reason.
Anti-Social Personality Disorder is completely intractable: meaning there is no known generally accepted treatment, short of incarceration in a penal facility, for any crime committed by someone with the condition.
Severe psychiatric complaints (schizophrenia, Bipolar Affective Disorder and Chronic Depression) are among the more intractable because they require lifelong medication compliance. That is, the only generally accepted treatment options are drug regimens, using drugs that have powerful side-effects. Patients on such regimens, once noticeable improvements have been seen, may discount past symptoms, focus on the side-effects, and decide they are "cured." This leads to non-compliance and relapse of symptoms.
Geriatric Psychiatric complaints are also notoriously intractable. Alzheimer's type Dementia, must be caught early if there is any hope of treatment, it is degenerative, and requires medication compliance similar to slow the development of symptoms. Medication compliance without a dedicated caregiver is notoriously poor, because deficits to short term memory, caused by the condition itself, often make remembering to take the medication difficult
A variety of other psychological and behavioral disorders are difficult to treat for a variety of reasons. Included on this list are: Borderline Personality Disorder, Anorexia Nervosa, Post Traumatic Stress Disorder, etc.