American Academy of Immortals
There appears to have been several groups to claim the title "American Academy of Immortals."
As a Generic Term
An 'academy of immortals' is essentially a who's-who of the greatest in a field; it could be a list of best-selling authors, Nobel-prize winners, record-breakers, etc., etc. It does not indicate literal immortality, but the immortality of their name by virtue of their accomplishments.
Academy of Immortals Worldwide
Academies of the Immortals existed, formally and informally, throughout cultured Europe. Their memberships included the greatest of artists, writers, musicians, and other creative high-brow minds. These Immortals are usually crowned by other immortals, or by a governing body of artists such as the French Academy.
Hosts of Philistia
The 'hosts of Philistia' is an artistic metaphor compared to the 'unwashed masses.' As a wordy way of referring to a group of people as 'philistines,' the host of Philistia were the antipode of the Academy of Immortals. As example of the usage:
After the French Government bought (the painting Whistler's Mother) for the Luxembourg it was thought, even by the hosts of Philistia, to be Whistler's best effort..."
Meaning that even the uncultured public found Whistler's painting of his mother to be a worthy piece of art.
This is the Academy I am most familiar with, through Hubbard's advertisments in his own The Philistine. These advertisements (from at least as early as 1900) promise, "on receipt of 10 dollars to pay for a Life Membership in the American Academy of Immortals, we record the new member's name on the Great Roster (in colors) and send gratis, express paid, one each of every volume of the "Philistine" which we have." It seems more a membership plan for early notice of newly published books; as a salesman, Hubbard was ahead of his time, understanding that fans deserve to be treated as such, and will even pay for the honor. He also encouraged members of his Academy to deliver the names of "boorish cranks," meaning "people who love beautiful things (not always knowing why) and do a bit of thinking on their own account." These friends of members would be sent a sample copy of the Philistine to peruse, and possible subscribe themselves. Hubbard's appropriation of the term "Philistine," meaning uncultured and pedestrian, and the admission of anybody for the lifetime enrollment of $10, is meant to be ironic and humorous.
This version, from a humorous editorial in the New Yorker from July 22, 1906 by Richard Barry, recounts a fictional gentlemen's club whose members are elected to the American Academy of Immortals -- claiming to be of "the parent organization, not the renegade Elbert Hubbard cult". The members are claimed to be the greatest masters in modern art. Again, the term is used ironically, to indicate a haughty self-worship worthy of referring to onesself as an 'immortal'.
A Line-o'-Type Or Two
Bert Leston Taylor, Chicago Tribune writer, created an "Academy of Immortals" in his column, A Line-o'-Type Or Two, in which people with strange, lurid, or sing-songy names are added to the roster.