It has been several weeks since it was announced that Jeremy Davies, best known for his role as Dr. Daniel Faraday on Lost, would be playing the main villain of upcoming Arrowverse crossover Elseworlds. Davies is a character actor known for playing a number of mentally unbalanced and/or socially awkward characters, and his Elseworlds character is no exception. Dr. John Deegan is described as being a psychiatrist at Arkham Asylum For The Criminally Insane, who is just as crazy as the patients he is supposed to be treating.
There is no pre-existing DC Comics character named Dr. John Deegan, which could just mean that Davies' mad doctor is a wholly original invention... but that seems like an odd choice, given that there's no shortage of mad doctors in DC canon to choose from. A more likely explanation is that Davies is playing a familiar villain, but his character's name has been changed for the sake of a reveal. If that's the case, our first likely suspect is Dr. Hugo Strange.
Related: Arrowverse 2018 Crossover Event Officially Titled Elseworlds
First appearing in Detective Comics #36 in 1940, Professor Strange (as he is often called today to avoid confusion with that other Doctor Strange) is one of Batman's oldest and most formidable foes. A master of psychology, chemistry and biology, Strange is a former employee of Arkham Asylum and one of the few villains who has learned that Bruce Wayne is Batman, using that knowledge to administer a series of tests of The Batman's skills. However, while Professor Strange is a formidable enemy, it seems unlikely that he could manage the likes of Supergirl and The Flash simultaneously. Also, the challenges he presents his enemies are mainly psychological in nature, which would be poor fare for a three-part, action-packed crossover event.
It is in considering that factor - the ability to handle multiple metahumans at once - that another answer suggests itself: Doctor Destiny. First appearing in Justice League of America #5 in 1961, Dr. John Dee was a scientist turned super-villain who built a number of devices to empower him in committing robberies. One of these devices, The Materioptikon, allowed him to both reshape reality from the fabric of dreams and control the dreams of others. He is perhaps most famous for his appearance in the two-part Justice League animated series story "Only A Dream", where he was able to successfully fight Superman, The Flash, Green Lantern and Hawkgirl at the same time by trapping them in their worst nightmares.
Given that level of power, Doctor Destiny seems a much more likely prospect for our mystery Elseworlds villain. Doctor John Dee, while never portrayed as a psychiatrist in the comics, has a long association with Arkham Asylum. There's also not much difference between Dr. John Dee and Dr. John Deegan - the name given for Jeremy Davies' character in Elseworlds. Time will tell the truth of it, but if pushed to make a guess, it seems fated for our heroes to face Doctor Destiny later this year.