Turns out, a surprising percentage of the DC Universe does. Setting aside characters who will not die of old age as a subset of their powers (Superman, Manhunter, Wonder Woman, Plastic Man, Black Adam, etc), there are still quite a number of people whose main schtick is immortality running around the DC Universe.
And most of them, it seems, are villains.
This makes sense to me. If comics are aimed at youth, then certainly the enemy should be age? Ra's al Ghul, Vandal Savage, General Immortus, and Felix Faust are pretty much grampas it's okay to punch. In contrast to heroes who are ageless due to their vitality, these guys won't shut up about how old they are. "I was conquering nations while your ancestors were still living in bogs!" is a line you could see each of these four saying. "I don't have to beat you, Luthor, I'll just out live you," is a line you probably wouldn't hear Superman say (even though that'd be a hella burn).
It's always a bit weird to me that these guys would want to come into conflict with superheroes in the first place. After the first thousand years of life or so, shouldn't you have moved beyond that whole "rule the world" instinct? I mean, if you're a few millennia old and your ambition is to rule the world, not having done so at least once is not exactly a great indicator as to your competence. "I'll get to it tomorrow," sighs Felix Faust as he opens up another tub of Sorcerer Chow.
One possible reason why your immortal ass has not move from couch to Throne of the World is that there are a bunch of other immortal assess out there gunning for the same seat. Thinking about it, that's probably a good thing. Without some conflict in your life that you can't simply wait out, forever must get pretty boring. Maybe that's why these four seem to go out of their way to clash with superheroes? Just for the novelty of it.
Anyways, it seems to me that immortals with a vested interest in the fate of humanity/Earth would not always be bad guys. There are, after all, some good guy immortals in the DC universe - Merlin, Shazam, Mitch Shelley, and Doctor Mist. Both good and bad immortals would want to, say, keep Earth safe from an alien invasion. While the bad guys might want to keep humanity penned up under their control, that at least implies that they want to keep us normals around.
Another common thread between immortals seems to be the use of agents and secret societies and the like. Ra's has his League of Shadows, Vandal has the Illuminati, and even the wizard Shazam operates through the agency of Captain Marvel. Folks like Doctor Mist who don't have a few go-fers hanging about them are the exception, not the rule (although I guess it could be argued that the Global Guardians was his). If you are a fan of conspiracy theories like me, it doesn't take long to realize that with all these secret and not-so-secret organizations running around that you'd end up with a cultist under every rock, a man in black lurking in every closet. That could get overcrowded, pretty fast.
So let's say that the immortals of the Drewniverse are at least somewhat organized. There is an elite club as it were for folks who will live forever. We'll call it the College of Immortals and membership comes with certain responsibilities. First, if you are going to be an active player in The Game, there are certain rules you have to follow (don't blow up the world being #1). Second, you are free to not play The Game, but that means you can't meddle in what's going on - sort of like a snowball fight where bystanders become combatants the moment they pack a handful of snow. This allows some immortals to free themselves of The Game and go off and do their own thing (Merlin would be an example of this - he's out of play at the moment, visiting dreamland from his stasis in a tree). There are probably some rules as to membership - you need to be at least X years old, born on Earth of at least half human parentage, etc. - that would rule out some of our more active immortals for membership (Superman and Manhunter ain't from 'round here, Wonder Woman was made not born, Mans Plastic and Resurrection still need to be around in another 150 years before they get nominated for junior membership, and so on).
Immortals who do not join the College would be crushed (we're talking chained up and thrown into a deep underwater trench here). Forever is simply too important to have unlicensed immortals running around. Immortals cannot strike at each other directly - Ra's and Vandal will not be brawling on the roof of Notre Dame, sorry. But their agents can, so League of Shadows vs Illuminati Men in Black squaring off against each other in the alleyways of Gotham? You bet. Heck, while they do that, I wouldn't be surprised in their bosses are sitting together in some five star restaurant reminiscing about the old days and what a doof Hannibal was.
So what's the motivation here? One-upsmanship for a start. Also, and this is probably more important, each member of the College wants to gain enough power to be able to betray the College and do away with it. Even the Good Guys would rather there not be a College out there encouraging evil dudes to visit pain upon an unknowing humanity - they'd rather topple the organization and get rid of the evil dudes. This constant conflict creates a nice level of stasis on the part of the immortals and encourages them to reach out to newer heroes and villains in an attempt to get an upper hand over their ageless rivals.
I wonder if there is a title in all this? I suspect there is, but I also suspect that it's been done, called 100 Bullets, and won a bunch of awards. Pity.
Instead, let's go with Resurrection Man (#36/52), even though that's against my better judgement - Dan Abnett should be writing more Gaunt's Ghosts books, dammit. We'll keep Mitch mostly as envisioned, with the major modification being his age. Rather than just now finding out that everytime he dies he's promptly resurrected with a new powerset, we'll have him not only aware of the power, but shirking it. Come Issue #1, he'll just have become old enough to qualify for membership in the College. He picked up his resurrection habit sometime in the period when Gentleman Ghost was neither a ghost nor a gentleman, but has been avoiding using it for some time, hoping that if he dies of old age, he'll die for keeps.
No such luck. When he opens his eyes on his death bed, having passed away at the ripe old age of 126, there's Vandal Savage looming above him, smiling, in his hand a parchment with Mitchell Shelley's name already written on it. "Welcome to your forever," Savage says, earning an enemy that will haunt him through out time. Abnett and Lanning can then take it from there.