(coll.), not all there (coll.), off one's head (coll.), off one's rocker (coll.), wrong in the upper storey (coll.), having bats in one's belfry (coll.), crazy as a bedbug (coll.), cuckoo (sl.), nutty (sl.), off one's nut (sl.), loony (sl.), a mental case, a mental defective, etc.
A clinic for such patients can also be discreetly referred to as, for instance, an asylum, sanitarium, sanatorium, (mental) institution, and, less discreetly, as a nut house (sl.), booby hatch (sl.), loony bin (sl.), etc.
In the story by Evelyn Waugh "Mr. Loveday's Little Outing" a clinic of this kind, treating only very rich patients, is described as large private grounds suitable for the charge of nervous or difficult cases. This is certainly the peak of euphemistic "delicacy".
The great number of humorous substitutes found in such groups of words prove particularly tempting for writers who use them for comical purposes. The following extracts from a children's book by R. Dahl are, probably, not in the best of taste, but they demonstrate the range of colloquial and slang substitutes for the word mad.
"He's gone off his rocker!" shouted one of the fathers, aghast, and the other parents joined in the chorus of frightened shouting.
"He's crazy!" they shouted.