It is possible to offer any of several ironic interpretations as to the intended meaning of the title of the book The Great Gatsby.
The person who started life as James Gatz reinvented himself as Jay Gatsby because he couldn't face the possibility of living the life he would have inherited if he had remained in North Dakota with his parents. Gatsby had a "Platonic conception of himself" which certainly indicated he thought of himself as being great.
Reality, however, showed Gatsby to be something less than great. Gatsby chased dreams that were never to be realized and refused to admit that inevitability.
his dream must have seemed so close that he could hardly fail to grasp it. He did not know that it was already behind him, somewhere back in that vast obscurity beyond the city,
Gatsby partially achieved the Great American Dream. To the extent that he attained financial wealth and entertained the high society of New York, he appeared to be great - but it was all an illusion, based on impressions. Gatsby was a great dreamer; he was a great pretender; but he was a great failure at achieving the ultimate dream of his life.