Caligula wasn't crazy. I agree he wasn't especially well adjusted as an individual, but then the Romans were often colourful characters. What we can easily observe is his immaturity. He takes nothing seriously except his own importance and safety. He plays games with people, he acts out roles, such as general, auctioneer, gladiator, statesman, and so on. However his sense of humour is black, and as a cruel personality, almost like a child torturing ants, he is callous to lesser individuals. In fact his sense of humour did nothing for his survival chances. Cassius Chaerea, the Praetorian Prefect at the time, was a war hero from the conflicts in Germania. Unfortunately, Chaerea also had a soft voice, and Caligula teased him mercilessly about his manhood. Right up until Chaerea - alongside other conspirators - stabbed him in a tunnel leading to the theatre.
There a re a number of anecdotes that seem to portray him as a nutter. Making his horse Incitatus a senator? He threatened to, on the grounds that the Senate were useless and his horse could do a better job, but anti-Caligulan propaganda and misinterpretation by witrnesses gave birth to the stories of his madness. Did he want to be worshipped as a god? Apparently, but then, this was considered normal for rulers in Egypt and it is no surprise he considered moving his capital to Alexandria (Egypt was also forbidden for Senators by Augustan rules thus he would rule as a god-king with no interference from those pesky layabouts in the Senate). His booty gathered at the expense of Neptune? Caligula had raised three legions to invade Britain, his attempt at military credibility, which like everything else he could not take seriously, staging fun and games along the way to the continental coast, where his troops, fearful of what lay ahead, mutinied on the beach and refused to embark. Caligula shamed them by ordering them to collect seashells - if Neptune was his enemy, then his booty will be from him, and presented the collection in Rome in order that the legions would be humiliated but of course the whole point was lost.
There's a great deal of speculation about all sorts of weird and wonderful ailments he suffered during his famous bout of illness but really that's like trying to diagnose disease from a story. Can you identify the mental illness suffered by Frodo as he bore the Ring toward Mount Doom? Any conclusion is possible.