Nemesis: Martian Manhunter
Other Major Foes: Superman, Batman
Appearances: Four comics.
Powers: Presumably the typical abilities of a Martian.
R’es Eda was the Keeper of the Sacred Martian Symbols on Mars II, and the best friend of J'onn J'onzz. However, R'es Eda believed that conquest was essential to Martian survival, so he faked his own assassination, framed J'onzz as a traitor, then resurfaced to lead an army on a genocidal path against an alien civilization. As it turned out, breaching their city would have killed all the Martians due to environmental contaminates, but Rʾes Eda didn't make it that far, after being exposed by his dying lieutenant N'or Cott. His bid halted by Martian Manhunter, Superman, and Batman, R'es Eda was locked up by his own people.
Lose: B'enn B'urnzz (0-9), N'orr Cott (4-6)
Ra's al Ghul
Other Major Foes: Batman Family.
Appearances: 400+ comics, extensive use in cartoons and video games, toys, and a major motion picture.
Powers: Excellent combatant with a prolonged life and legions of minions.
Bio: Ra's al Ghul believes that the only way to save the world is by eradicating most of humanity. With his centuries long life span enabled by the Lazarus Pits, Raʾs al-Ġūl has accumulated knowledge and martial skills to come close to his goal time and again, if not for the intervention of the Batman.
It would be easy to dismiss Rʾes Eda against one of the most famous and popular comic book villains, but hear me out. Raʾs al-Ġūl is the head of an international criminal organization, the League of Assassins, has a personal army of ninja Man-Bats, plus his daughter is Talia and his grandson is Damien Wayne. With all of these resources at his disposal, al Ghul spent over twenty years getting beaten by Batman and Robin, before moving up in the world and getting beaten by the Dynamic Duo and their extended family of non-powered urban vigilantes. Remember that one time Ra's stole Batman's personal playbook to defeat the entire JLA, and still failed inside of a four issue arc without ever doing anything remotely as cool in the thirteen years since?
R’es Eda had an army of Martians that repelled Superman and Martian Manhunter! Sure, R’es's plots are nearly as convoluted and illogical as Cay'an's, and he wears a silver swan helmet with pink streamers that do not match his man-skirt. He still managed to drag Supergirl, Hawkman, Batman, and Hawkgirl through his muck! We can't assume R’es Eda had the full compliment of Martian super powers because he didn't display any, but he also was never shown to have the vulnerability to fire, either. I'm not guaranteeing a Rʾes Eda win, but he worked on a bigger scale than Raʾs al-Ġūl, has more inherent abilities, and has fended off a higher caliber of foe than the Demon's Head is accustomed to in his one extended story arc.