The Batman identity itself is one. In DC's main universe, there have been four people to take on the mantle of Batman after Bruce Wayne.
After being crippled by Bane during the Knightfall storyline in The '90s, Bruce leaves the mantle to Azrael (Jean-Paul Valley). Azbats eventually lost his shit, and Bruce offered the mantle to Dick Grayson. Dick Grayson took up the mantle on what both knew was a temporary basis.
After Bruce was presumed dead following Batman R.I.P. and Final Crisis in 2009, a big deal is made out of who will succeed Bruce. Dick Grayson doesn't want to take on the mantle, which leads Jason Todd and Tim Drake to fight over which one of them will do it. Eventually, Dick comes around and became the new Batman, with Damian Wayne serving as his Robin. Once Bruce returns, both acted as Batman before Dick went back to being Nightwing.
After, after Bruce was rendered amnesiac in the aftermath of the 2014 Batman: Endgame storyline, James Gordon took on the Batman mantle to protect Gotham in the hero's absence as an officially sanctioned GCPD Batman. Dick had agreed to take up the mantle again in the same story, but became preoccupied with other things.
In 2021's I Am Batman, Timothy "Jace" Fox, the son of Lucius Fox, became the fifth Batman. Unlike Bruce's previous successors, when Jace initially takes up the cowl, Bruce was still active as Batman. Jace and DC's promotional material distinguish the two by still referring to Jace as Batman, but referring to Bruce as the Batman.
The Batman identity has also been taken over by a multitude of people across DC's various alternate futures and elseworlds.
DC One Million has a grieving man take up the Batman identity in the 853rd century, after the death of his family. He's the warden of the prison planet of Pluto.
In at least one alternate future, Tim Drake himself will become Batman. He's a murderous version, however, and when the main Tim Drake meets him, he resolves to never become this version of himself. A later story did some Arc Welding to tie this story in with the rest of the Robins, establishing that Tim becomes Batman after Dick retires as Batman and Damian fails to save Gotham (with the implication that Tim killed Damian).
In another Bad Future, Damian Wayne will become Batman after Bruce retires and Dick is killed.
Canon Immigrant Terry McGinnis actually takes up the Batman identity after Damian in that same story, Damian acting as his mentor. The story establishes that the Batman legacy will be carried on for centuries.
In yet anotherBad Future, Terry inherited the title from Bruce, and got sent back in time to Set Right What Once Went Wrong, ending up five years into the future of the main DCU. He died in the attempt, and the Tim Drake of that future took up the mantle, travelling to the present of the main DCU to complete Terry's mission, then getting thrown to a now-changed future, where Terry was missing and Tim temporarily took his place until he was found. Ow.
The Elseworlds story Dark Knight Dynasty reinvented Batman as a Legacy Character, creating both an ancestor of Bruce's called Sir Joshua of Wainwright who wore bat-motif armour, and a descendant named Brenna Wayne who became a 25th century Batwoman. All three of them faced Vandal Savage.
In Earth 2, Bruce-as-Batman dies in the first issue; much later, another Batman emerges, who turns out to be Bruce's father Thomas Wayne, who secretly survived the mob hit that killed his wife. He in turn is replaced by Earth-2's Dick Grayson, who, after an incident that keeps him from being Batman, is replaced by a woman, Helena Wayne, the first Robin of this universe and Huntress. She just wears a heavy coat and pretends to be a man.
DC Future State takes place some time in the future across its various series. For the Batman titles, Bruce's Batman is presumed dead and is working completely in secret. In his stead, a new character, Tim "Jace" Fox, has taken up the Batman identity. In this timeline, the previous Batmen were Bruce, Azrael, Dick and Jim, making Jace the fifth Batman.
In the Silver Age, it was established that Bruce himself had taken up a Batman legacy begun by Thomas Wayne, and that Bruce was the original Robin during his years of training to become Batman. Both of these were quietly discarded post-Crisis.
Actually defied in the Silver Age. In the Justice Society of America two-parter "Only Legends Live Forever", the Golden Age Batman dies trying to apprehend a super powered villain. After the funeral, Dick Grayson decides on the best way to keep Bruce's memory alive. Helena Wayne, Bruce's daughter and the pre-Crisis Huntress, cries out "No!" and it seems that both Dick and Helena wanted to take over the role of Batman. However, she had the same idea Dick had - Batman's too big to be just a man in a costume, especially with his identity revealed, thus they decide to honor his legacy by continuing on in their own identities as Robin and Huntress.
Batman: Two Faces, an Elseworld title consisting of an adaptation of The Strange Case of Dr. Jekyll and Mr. Hyde where Bruce Wayne is Jekyll and the Joker is Hyde, ends with Bruce Wayne begging Harvey Dent to let him die and to take the serum that was intended to cure Harvey of his psychosis. After finally regaining his sanity, Harvey Dent also chooses to honor Bruce's memory by becoming the new Batman.
The Robin Hood inspired identity of Robin is a somewhat special case, each having been the sidekick to the same character (though, as noted above, he's also a legacy character, and two of them served under other Batmans in addition to Bruce). Each eventually took on a different identity after their time as Robin.
Dick Grayson, who went on to become Nightwing, left the job to Jason Todd. Dick Grayson also served as Batman (see above).
Pre-Crisis, the first Nightwing was actually none other than Superman himself, Kal-El/Clark Kent, who needed to become a Batman-like vigilante when he was in the Bottle City of Kandor. He later allowed his cousin Van-Zee (who had an Uncanny Family Resemblance to the Man of Steel) to use the identity to fight crime in the bottle city after Superman left. The best known Nightwing, former Robin Dick Grayson, then decided to adopt the name after he left Batman as a homage to both him and Superman.
Post-Crisis, Nightwing was the name of a Kryptonian vigilante that Supeman told Dick about, and Dick, being inspired by the story, took up the Nightwing identity. After One Year Later another former Robin, Jason Todd, became a more sociopathic version to mock Grayson, and a woman named Cheyenne Freemont became a heroic version to help Dick. When Power Girl went to Kandor she also adopted the Nightwing persona. The mantle was briefly passed to Chris Kent, the adopted son of Superman. Sometime after, poor Chris was trapped (seemingly forever) in the Phantom Zone. Dick Grayson retook the Nightwing identity following his lengthy tenure as Batman, although Chris had been RetGone'd by that point.
Nightwing’s sidekick Flamebird was originally none other than Mr. Action himself, Jimmy Olsen. After he left the bottle city the former Elastic Lad gave the title to the new Nightwing's son-in-law Ak-Var. Supergirl took the identity when she went to Kandor. Even though the first two were male the name is now associated with female characters so much that when Superman retook the Nightwing identity, Lois Lane and not Olsen was his sidekick, and the sixth Flamebird, Thara Var, is the wife of the second Flamebird (who apparently no longer exists). Thara was eventually killed.
Post-Crisis, the original Bat-Girl Bette Kane became Flamebird to erase her time as Bat-Girl and to contrast against Dick Grayson, to the point where her main goal during the 2000s is to prove that she's a competent vigilante in spite of Nightwing.
Bette: ...Quit? ...like hell I will.
Jason Todd was killed by The Joker. Jason Todd stole the Red Hood identity from The Joker (making him a Legacy Villain) and also wore the costume of Red Robin, but returned to his Red Hood identity.
The Killing Joke depicts the Red Hood identity as a sort of extra-disposable legacy character shared by the members of a criminal gang, who foisted the hood on whichever accomplice they'd recruited for a specific job and didn't want to risk being seen with. Whether that's right or not depends on which Multiple-Choice Past you prefer for The Joker.
In the New 52, the Red Hoods were a street gang all wearing the same mask, and the Joker was their leader, going by "Red Hood One". He himself usurped the role from the previous leader. Batman speculates that it's unlikely that one was even the first.
Tim Drake figured out Dick and Bruce's identities, and when he went to persuade Dick to become Robin again he found himself stepping into the position. Later on, after Damian Wayne usurped the role, he would become Red Robin. In the New 52, it's said that he was never Robin, and was always Red Robin, however his original origin was restored in DC Rebirth.
The Red Robin identity is also a legacy character. Essentially, an Elseworlds Dick Grayson wore it and passed it on to the mainstream Jason Todd. Jason wore it for a while but ditched it, and Tim Drake picked it up after being replaced by Damian Wayne.
At one point, Tim had a 10-Minute Retirement from being Robin, leaving the space open for Stephanie Brown, formerly a solo heroine known as the Spoiler. She then died in a Story Arc that shook up Gotham City and readers pretty badly and caused Tim to return to the job. She would go on to become the third official Batgirl.
The fifth Robin is Damian Wayne, Bruce Wayne's son. In this case however, he started off as Robin to Dick Grayson while he was Batman, before becoming Bruce's new sidekick once Dick went back to being Nightwing after Flashpoint. After killing Brother Blood in Teen Titans (Rebirth), however, Damian rescinds the Robin mantle, believing he failed his father by violating his principles. This led to Tim Drake (who had adopted a new superhero identity as Drake in Young Justice (2019) not long before this storyline) becoming Robin again.
Extra-Canonical Bad Future Robin Carrie Kelley appeared in Batman: The Dark Knight Returns. She was inspired to become Robin by Batman's return from retirement and met up with Bruce by chance. In this universe, Dick Grayson was the first Robin and Jason was the second. She is now a Canon Immigrant, but only briefly wore the Robin costume for a Halloween party.
After Damian's resurrection and Bruce's death, Damian would leave Gotham to go on his own adventures. In his absence, various youths took the Robin symbol as their own, and they all function as a group with the Robin name. None of them work with Batman or even the rest of the Bat-family though, nor has any one member of the group referred to him or herself as Robin.
Bruce Wayne himself also spent some time as Robin, thanks to some Silver Age silliness in Detective Comics #266 (which is also referenced in the classic The Untold Legend of the Batman). Essentially, Bruce learned all he needed to learn to become the World's Greatest Detective from a detective named Harvey Harris. Harris himself decided on the name of Robin for the disguised Bruce, as the costume reminded him of a red-breasted Robin. The reason for the disguise is that Bruce didn't want Harris to try and dissuade him from fighting crime. You'd think that Bruce would have realized this wouldn't really stop him, all things considered... but then again, Silver Age. As you might guess, this last one was removed from continuity with the "Crisis on Infinite Earths", and certainly no longer exists after Flashpoint.
In Earth 2, the first Robin was Helena Wayne, the daughter of Bruce Wayne and Selina Kyle. She disappeared from Earth 2 the day her father died, and ended up on the mainstream DCU and called herself Huntress. She eventually takes up the identity of Batman back on Earth 2, and her Robin is Dick Grayson's son.
The whole trope is discussed and deconstructed in one issue of the Robin solo series. The issue is told from the perspective of the Joker as he reflects on his relationship with Batman and Robin while plotting his next scheme from Arkham Asylum. Joker has connected the dots that there have been multiple Robins, as he knows for sure he killed at least one (Jason Todd) and the current one (Tim Drake) is discernibly younger than the first one Joker encountered (Dick Grayson). Based on this, Joker finds himself wondering just how many Robins there have been, then further wondering if there have been multiple Batmen too and if he really did kill the Caped Crusader in some of their previous battles, in a bizarre inversion of Joker Immunity.
It's similarly lampshaded and discussed in Gotham Central when a teenage boy in a Robin costume turns up dead. Since it's pretty much public knowledge by this point that there are more than one Robins, the cops and public both find themselves wondering if it's the real one or not, and the kid's parents publicly state that they think their son may well have been a Robin (he's too young to have been the original; they believe he's one of the replacements). The Teen Titans end up having to stop by GCPD headquarters with Robin in tow just to make clear that, no, the victim is not the real deal.
Betty Kane was the original Bat-Girl introduced in the 1960s. Whilst only having a few appearances in the Bat comics themselves, Betty would become a more Teen Titans-afilliated character, becoming a member of the short-lived Titans West as well as attending Donna Troy's wedding. After Crisis occurred, she would now be called Bette Kane and her time with the mantle of "Bat-Girl" was erased, instead now calling herself Flamebird, similar to the Kryptonian legend of Nightwing and Flamebird. However, during her brief appearance as part of the New Young Justice, Bette fought alongside Cassandra Cain and had her previous history restored.
Bette: (to Cass) Batgirl, huh? Been there, done that.
Barbara Gordon became Batgirl, the "first" Batgirl in current continuity since Betty Kane's time was (temporarily) erased, and held the role until The Killing Joke, when The Joker shot and crippled her.
The first actual legacy transfer occurred when the Batgirl mantle was picked up years later by Helena Bertinelli, the Huntress, when she realized the crooks were more afraid of Batman than they were of her. This only lasted a short while however, since Batman never gave her his consent or blessing.
Eventually the costume passed to Cassandra Cain, an "official" Batgirl with the consent of Barbara and partnership of Batman.
Cassandra unceremoniously gave up the identity in the Batgirl (2009) series and took on the new moniker of Blackbat. The cowl was picked up by Stephanie Brown, previously the Spoiler and the fourth Robin.
In Post-Flashpoint continuity, Babs, Cass and Steph became Batgirl, the second Orphan -taking over the identity from her father- and Spoiler. In the 2022 Batgirl series, though, they agreed to share the Batgirl title.
Batman and Robin's British counterparts aren't exempt, either. The original Knight and Squire were the Earl of Wordenshire and his son Cyril. The Earl was killed by one of his archenemies, and Cyril sank into depression, ending up in the gutter, where he was found by Beryl Hutchinson. Cyril took up his father's code name, becoming the new Knight, and Beryl became the new Squire. After Cyril’s death, Beryl became the new Knight.
Clayface, one of the Batman villains, has had no fewer than eight entirely different people take the name and powers. (Most of them have shapeshifting powers, and most have an additional power which is different for each individual.) In an odd divergence, the original Clayface didn't actually have any powers — until he arranged to duplicate those of his three "heirs".
This trope is played with in the beginning of Gotham Central, dealing with the actions of Firebug, an arsonist villain who has just returned to Gotham after a long absence. The Gotham City Police Department eventually managed to track down Joseph Rigger and arrest him after him tried to escape, only for him to reveal that he sold his costume two years ago after it almost killed him. The real culprit is deduced and arrested, and it turns out to be an upper-class twit who bought the suit and runs around burning down banks for the adrenaline rush.
There have been several Two-Faces,including the first Two-Face Harvey Dent, his butler Wilkins, Paul Sloane, an actor, George Blake, a petty criminal and Harvey Apollo, another actor. An averted case with the New 52 Two-Face. There's only ever been one, Harvey Dent. But a story arc called "The Big Burn" was seemingly intended as the definitive ending for Harvey, with him committing suicide at the end of it. However, during the story, the person who made him Two-Face, Erin McKillen, gets scratches on one side of her face and becomes leader of the Blackgate inmates, which sets her up as the new Two-Face. However, Scott Snyder would bring back Harvey for his first arc of All-Star Batman, and Erin entered Comic Book Limbo.
The Azrael identity is this, a line of assassin-enforcers for "The Sacred Order of Saint Dumas". It initially belonged to Jean-Paul Valley (who'd inherited it from his father), but he discarded it after becoming Batman. Years down the line it was picked up by Michael Lane, a cop conditioned so that he would replace Batman should the need ever arise. At the beginning of the New 52 reboot, Lane was still Azrael, but years down the line, Jean-Paul Valley was reintroduced (seemingly never having been Batman), and is apparently the first Azrael.
Batwing was an identity first used by David Zavimbe, a former child soldier and the Batman of Africa under the Batman Incorporated initiative set up by Bruce Wayne. He eventually quit, and the Batwing identity was taken up by Luke Fox, the son of Bruce's associate Lucius Fox. Since the change, Batwing gradually became a full fledged member of the Batfamily.
Batwoman: The first Bat-Woman was Kathy Kane, a billionaire heiress. The character never caught on and seemingly never existed post-Crisis. But then Infinite Crisis happened, and in the pages of 52, Kate Kane takes up the identity of Batwoman, and is seemingly the first... then Batman (Grant Morrison) reincorporated several parts of pre-Crisis canon, including Kathy Kane having been Bat-Woman, meaning Kate is the second. There is no relation between the two characters, despite them having the exact same name, and the New 52 would change Kathy's name to Katrina Netz, with Kathy Kane being an alias.
Probably the craziest example in The DCU is The Joker's Daughter. Appearing in the mid-70s Batman titles, she claimed to be the daughter of a different Bat-villain each time, with a costume to match. She was eventually revealed as Two-Face's daughter, who rejected her and her mother because he wanted twins. She then joined the cast of Teen Titans as "Harlequin" until that title's cancellation not long after. Afterwards, she mostly disappeared, cameoing Post-Crisis as a mental patient and staying that way until the run-up to Infinite Crisis, which had her gleefully claiming to be the daughter of everyone from Doomsday to Punch and Jewelee, until she was killed for shock value and retroactively revealed as the daughter of the Mirror Universe heroic version of The Joker, the Jokester.
In Death of the Family, it turns out Harleen Quinzel isn't the first Harley Quinn. She is, however, the first one to escape with her life.
In Batman Annual #20, a Batman cyborg fights the City Controllers who rule New Gotham through oppression and mind control.
In Batman: Shadow of the Bat Annual #4, the City Hero, the Batman, is the guardian of Nu-Gotham who fights crime in the city and protects it from external threats such as the Lizard-Men. The people of Nu-Gotham believe that the Batman is their ruler King Bruce the 23rd but it is in fact his retainer Richard Grayson as the King was too frightened.
In Batman: Legends of the Dark Knight Annual #6, successive generations of the Kane family, calling themselves either Batman or Batwoman, have been the executioners of the city of Gotham since their ship, the Plight of Gotham, crashlanded 500 years earlier.
In Robin (1993) Annual #5, a former proctor on the Generation Ship Gotham who had grown disillusioned with his society was inspired by the legends of Old Earth to become the new Batman. A rebel named Tris Plover becomes his Robin.
Grant Morrison made two members of the Batmen of All Nations into legacies. In JLA (1997), Percy Sheldrake, the Knight of Britain, is revealed to have been killed by his archenemy Springheeled Jack, and his role taken by his son Cyril, formerly the Squire. His own squire is Beryl Hutchinson, who would later take the role of the Knight after Cyril's death at the hands of Leviathan in Batman (Grant Morrison), and later get her own Squire. Earlier in Morrison's Batman, the Ranger of Australia was murdered by his teammate Wingman, and the mantle was taken by his previously unmentioned sidekick Johnny Riley, the Scout.
