Lelouch Lamperouge from Code Geass mostly fits this trope, except that rather than pretending to be stupid, he pretends to be too Brilliant, but Lazy. It's not a difficult pretense, because that was precisely the case until he Jumped at the Call in the first episode.
Of course, that he constantly skips class in order to con money out of the rich and powerful by preying on their arrogance is the reason people think that he's lazy, (along with his great distaste for having to commit to physical labour) as he's quite clearly capable and willing to go to absurd lengths to manipulate events for his own whims, from world conquest to helping a little bird with a broken wing.
Kallen also somewhat falls into this category, as her father IS rich, and she spends nights fighting Britannia. Except replace "idiot" with "Ill Girl".
Jiraiya from Naruto qualifies. Ostensibly a wandering (and lecherous!) carefree hermit, he is actually quite wealthy...and in fact one of the Legendary Sannin.
Batman: Bruce Wayne. He used to use this all the time, but after a while, realised that it hurts his endeavours as Bruce Wayne, such as his attempts to fix Gotham, and has pretty much done away with it altogether. He now acts as a level-headed CEO and philanthropist who is just very secretive. What probably helps is that, he "reveals" to the media that he's been working ''with'' Batman, so that's what people assume he spends his time on.
The entire Club of Heroes is pretty much this; When invited to a reunion, Batman even says he's curious what happens to bored rich idiots when they're washed up. Chief Man-of-Bats flat out admits that whatever he is or was, he was never really a hero. The Musketeer also revels in the fact that his book made him a millionaire overnight and that he sold the movie rights for even more money, meaning he's internationally famous but never has to fight crime again. That's right, he got richer and also quit his night job.
He may secretly be the World's Greatest Detective, but I think that the average person in the DC Universe thinks of Bruce Wayne as the World's Worst Skier. - Tirian
The Batboys Dick Grayson, Tim Drake and Damian Wayne, have usually averted this, and are seen as being extremely intelligent by the general public. Damian in particular makes no effort to hide how smart he really is, and even ran Bruce's company while he was thought dead. Dick, meanwhile, has held a very unusual mix of jobs, including police officer, museum curator, gymnastics instructor, bartender, male model, and Waynetech executive. On the other hand, he has been called "Gotham's very own Paris Hilton", so it may be a case of Depending on the Writer.
Kate Kane, Batwoman, is also this. To most people she's a military washout who parties all night and sleeps all day with her dead mother's money. We see this hurting her social life early in her series - a girlfriend sees that Kate clearly hasn't slept and dumps her, assuming she's been out on the town with someone else.
Green Arrow: Oliver Queen, at least in some renditions. Before he became Green Arrow, it was said that Oliver Queen's sole survival skill was "making a martini last an entire hour".
Averted by Ted Kord/Blue Beetle II who is very involved with Kord Industries, an R&D firm.
PS238: The Revenant. He's one of the few that openly admits to gobs of cash in his costumed identity — as he commented (paraphrased), "Sometimes I think having access to (lots of) money is the greatest superpower." Amusingly subverted, as well, since he's wanted by the government ("warrants for arrest in eleven states"), not for being a vigilante, but because only one of his (many) identities pays taxes.
Averted with Iron Man. Anthony "Tony" Stark is well-known as a workaholic engineer and business owner/government minister who definitely earns his money.
In the comics, he also plays as hard as he works, but it tends to be between projects and/or a backdrop to high-level negotiations.
Alan Moore gave Dan Dreiberg, aka Nite Owl, this background, and the expensive toys that go with it, in Watchmen — although in contrast to the more usual Obfuscating Stupidity, his cover is that he's a harmless intellectual. Played with in that, as a crimefighter, Nite Owl is decisive and confident, while Dan Dreiberg is nervous and impotent. (contrast with Rorschach, who is poor and a lunatic, with no day job).
Watchmen originally was going to feature the Charlton Comics lineup, which includes Blue Beetle (the second — Ted Kord), who also fits this trope (though, like many of these examples, Blue Beetle is supposed to be a genius inventor within his setting, rather than a true "rich idiot").
Adrian Veidt was already a millionaire, albeit not an idiot (World's Smartest Man, in fact), and one can presume he used this as his cover. In an interesting variation, he became even more filthy rich after revealing his identity as Ozymandias and then starting a marketing business based on it.
Sandman Mystery Theatre: Wesley Dodds was a mild form of this. He was very geeky, no one thought he was dumb by any means, and he was shown to be actively involved in business ventures (although usually in the background of the story).
DC's recent reinvention of the old Archie Comics character The Web is one of these. The twist is that he's actually an inversion- whereas Batman is a grim vigilante who fakes being a feckless playboy, The Web is a feckless playboy who took up vigilanteism as a hobby. Then criminals killed his brother. In contrast, the original 1940s version was a college professor/criminologist who moonlighted as a vigilante, and the 60s version was a Henpecked Husband who had to sneak around his wife to fight crime. So more of a "comfortably well off intellectual with a job that leaves him plenty of free time" than this trope.
Ariana Von Holmberg of Rashida Jones' Frenemy of the State offers a female twist on this character as an Upper-Class Twit recruited by the CIA.
In Quantum and Woody, Woody is technically one of these; after the death of his father, he's inherited a fortune and a sizable stake in the company. However, the trope is subverted because a messy divorce settlement and several inheritor clauses means he has no direct access to his fortune, but must instead get regular payouts from the estate's executor... which happens to be his Vitriolic Best Bud Quantum.
Invoked in one half-issue of The Flash, when the second Trickster speculates that his Secret Identity is one of these—"I can tell he's one of those pretty boys under that mask. Probably has jet black hair. Dozens of girlfriends. Bet he lives in a mansion somewhere, too." He happens to be completely wrong, since the Flash at this point is a redheaded, Happily Married police mechanic. Doubled as a Mythology Gag regarding Bruce Wayne, of course.
A villainous variant is done in Runaways, where the parents of the six protagonists pretend to be your standard dull but well-meaning affluent, upper-class couples, when in reality, they are twelve incredibly corrupt and powerful super-villains.
Mystery Men: Wealthy lawyer Lance Hunt is the alter-ego of Captain Amazing. Since Amazing is very focused on making money through sponsorships, and isn't exactly the brightest guy around, he's probably a pretty lousy lawyer.
The Shadow: After the protagonist reforms and devotes himself to fighting crime, he adopts the cover identity of wealthy man-about-town Lamont Cranston. (To keep things simple, the movie avoids mentioning the complicated backstory from the pulps, wherein Lamont Cranston is an actual separate person whose identity the Shadow borrows. The novelization by James Luceno brings it back in, along with several other bits of pulp continuity the movie left out.)
In The Dark Knight Saga, Bruce acts as a well-meaning, but seemingly ditzy philanthropist, but keeps that hidden from the public by using Lucius Fox both as a decoy and an actual trusted advisor — probably to keep suspicions down.
In The Dark Knight, Bruce actually provides an interesting example of being A Serious Man, who pretends to be a Rich Idiot, pretending to be a Serious Businessman. The world is of the impression that Bruce Wayne is the dignified head of Wayne Enterprises, while those who actually meet him are of the impression that he is a Rich Idiot and Lucius Fox is the real brains of Wayne Enterprises. While a small selection of people, including Alfred and Lucius Fox, know who he really is.
Iron Man: In the movie, Tony Stark seemed to be more of a Rich Idiot with a day job, where he only had to pay attention as long as he didn't get bored. Naturally enough, that was almost his downfall. However, it should be noted that, in the first movie at least, he is perhaps the world's most brilliant engineer, but only barely competent as a business manager, mostly delegating that part of his job to his hypercompetent secretary, Pepper Potts, and his trusted lifelong friend, Obadiah Stane.
And at the end, he goes and renders the whole Secret Identity trope irrelevant by revealing the truth at a press conference.
The Shadow: Lamont Cranston. However, Walter Gibson later revealed that the Shadow only impersonated Lamont Cranston, a separate person, in order to move about in the circles of the wealthy for intelligence. Kent Allard, a former secret agent, represented the Shadow's true identity. The Shadow had access to Cranston's wealth due to Cranston having taken an extended tour abroad. In the meantime, Allard had begun to impersonate him and forged his signature (it also helped that the two were dead ringers for one another). Once Cranston returned to America, The Shadow gave him an offer — either go back abroad, allowing The Shadow to continue to use the Cranston persona, and gain a very tidy allowance for the rest of his life (Allard had been a very good, if unauthorized, steward of Cranston's money), or have The Shadow ruin him financially.
Francisco Domingo Carlos Andres Sebastián d'Anconia from Atlas Shrugged. Though he has a well-known day job as owner of d'Anconia Copper, he still uses his millionare playboy status as Obfuscating Stupidity. Unlike most rich idiots, the public persona that spends money like water and can't make a good business decision to save his life ends up driving d'Anconia Copper directly into the ground...All According to Plan.
Prince Diarmuid of The Fionavar Tapestry generally gives the impression of being a shallow, frivolous twit who gets away with his shenanigans (like sneaking into an enemy country to woo their princess) because he's the heir to the throne. It works, until you realise that most of the crap he pulls actually requires quite a lot of strength, bravery and cunning.
Alec Checkerfield from The Company Novels is an earl who plays at being an idiot, and on his own time funds various charities and countries while making money via being a pirate. Oh yeah, and he's a computer genius/cyborg who later decides to fight evil. At one point he gets compared to The Scarlet Pimpernel, but since it's the 24th century, he has no idea what that means.
Henry Fitzroy from the Blood Books (and Blood Ties TV show) is a (romance or comic, depending on the medium) writer by night, somewhat subverting the trope in that he actually does something for a living. He seems to be fairly financially well off, presumably because he's saved up money over the years.
American Psycho has a twisted variant: Patrick Bateman comes off to most of his acquaintances as a Rich Idiot With An Unnecessary Day Job (at least one of his girlfriends notes his dad owns the company and he's wealthy enough to not have to bother), but it's all part of a none-too-successful effort to fit in.
Ripliad is equally twisted: Tom Ripley spends his days lounging around in Mediterranean European countries, occasionally killing someone when he really has to. Considering the books pretty much codified the Villain Protagonist trope, it counts as a subversion.
Several Forgotten Realms novels by Elaine Cunningham feature Danilo Thann, a young nobleman from Waterdeep, inspired byThe Scarlet Pimpernel. In Elfshadow he initially appears as a none-too-bright dandy who dabbles in (frequently miscast) magic and (bad) music; Arilyn Moonblade briefly takes him hostage and ends up stuck with him, to her great dismay. It's eventually revealed, however, that Danilo is actually a member of the secret society of heroes known as the Harpers, generally very savvy, and was actually assigned to observe Arilyn all along. Another character who knows Danilo's true status observed that nobody would ever believe that he of all people is a Harper.
More than once Danilo is shown getting heartily sick of playing the idiot, but he reluctantly continues because he can't deny how useful a cover it is.
The first Deathstalker series featured as one of its side characters the son of a noble clan who was well-known for his sole skill of always being up to date with court fashion, no matter how extravagant or obscure. With everybody busy focusing on other more obvious threats than the well-dressed idiot, nobody figured out he was an assassin working for the rebellion for some time.
Kylar from The Night Angel Trilogy poses as a poor nobleman as his cover identity. In reality, he is an apprentice assassin. While Kylar does have obscene amounts of money, he can't use it since that would draw attention to him, but posing as a poor nobleman gives him both access to his targets and anonymity.
Seregil and later Alec play this in the Nightrunner series. Seregil is seen as an exile, a failure at the court, a party boy, a great listener with no strong opinions, and decadent rich idiot. In truth he is a master swordsman, famous cat burglar for the Nobles, and probably the greatest spy the kingdom has Alec is groomed as this as well though most dismiss him as Seregil's boytoy. They both find the act a burden and are eager to escape it.
Don't look at me. I'm a drunken, chemical-besotted playboy who does nothing but cavort, sleep, and feed. And even if I had the mind to take a bit of vengeance on the Red Court, I wouldn't have the backbone to actually stand up to anyone. I'm totally harmless.
Zorro - Don Diego de la Vega acts like a nebbishy bookworm to deflect suspicion that he is the titular vigilante.
While he's not a classic example, it's been impossible to tell how Peter Petrelli of Heroes pays for his groceries since he resigned from his nursing job halfway through season 1. The Petrellis are rich, but given that he's stated several times that he didn't want to live off his father's money, it's all a bit of a mystery. Hiro Nakamura is a straighter example of the trope, since he has apparently inherited money from his father.
Peter did get another job as a paramedic halfway through season 3 and he spent season 2 and the first half of season wrapped up in the plot so it's possible he might have been able to avoid this for a while.
Averted in the 1960s series The Green Hornet (and the original radio series), in that Britt Reid's day job is publishing The Daily Sentinel, a newspaper that's been owned by his family for years.
However, in the original radio series and in two film serials based on it, this trope gets some lip service: while Reid has a day job, he often ignores it much more than his family likes.
Tsurugi Kamishiro is this trope in the most literal sense, emphasis on "Idiot." But then he finds out his family fortune has long-since dried up, and gets an actual day job, and clings to the "idiot" part for dear life.
Lampshaded and subverted in Star Trek: Deep Space Nine. Julian Bashir likes holonovels in which he is a James Bond character. On one occasion Garak (a real secret agentcompletely innocuous tailor) joins him in one of these games. When Julian explains that his cover is a rich playboy, Garak remarks "I obviously joined the wrong secret service".
Sort of in Have Gun — Will Travel. Paladin plays the role of city dandy in San Francisco, and generally wears light colored clothing there, but when hired for a job, puts on an all-black ensemble. However, he calls himself Paladin in whatever location he's in, so there isn't a pure Secret Identity here.
The Ashwood Abbey society is a group of rich socialites who are so bored with life that the only thing that can provide a sufficient thrill is hunting down vampires and werewolves.
On the even more villainous end of the scale, we have the Hunt Club. They Serial Kill as a point-based sport.
In Champions, 15 points buys you the "Filthy Rich" Perquisite, which turns your character into this.
In GURPS there is a handwave/justification available for this in the form of the Advantage "Independent Income" that represents having "people" who will do some of your work for you (actually taking time out of your day to work is still required if you want your full income, though).
The classic character Danilo Thann is the very image of inane and decadent highborn dandy, mildly annoying due to his habit of singing bawdy couplets and hurling spells that tend to fail in spectacular ways. Behind the image is a Harper secret agent, a wizard who managed to impress his master Khelben "Blackstaff" (a champion and apprentice of the gods of magic) time and again and dares to confront him, a talented bard and a decent fencer. Even shrewd opponents disregarded him — until it was too late.
Haedrak III, the Tethyrian throne's last heir who participated in a war, has a Rightful King Returns celebration... and that's about it. Yet another useless lordling dabbling in magic and absent-mindedly playing with his familiar instead of listening to the serious discussions. Haedrak Errilam Alemander Olosar Lhorik was also previously known as "Lhaeo". Remember that Obstructive Bureaucrat with Photographic Memory who for more than twenty years was the Old Mage's assistant, Seen It All and survived to never tell about it? So, he was also an apprentice (and Elminster developed a handful of spells personally for him), knows almost all the important people on the continent and lots of very exclusive secrets. After crowning Haedrak proved to be an investigator good enough to let Khelben (having half of the Harpers to spy on his behalf) know some things happening in Blackstaff's own city. Of course, he didn't stop working with Harpers just because he doesn't serve one of their founders anymore.
Donnie Rhodes, Scion of Aphrodite, is a darker version of this — his father opted to give him as little attention as legally necessary (the elder Rhodes didn't take the whole "a goddess dumped their kid on his doorstep" thing well), so he threw himself into this lifestyle. His mother saved him from getting shot in a nightclub one night, and he grew out of it once he met his Band. Losing his Band in Scion: God caused him to revert a little.
The default setting for player characters in Spirit Of The Century, thanks to its pulp sensibilities. Even Centurions who aren't filthy rich can access the considerable resources of the Century Club to maintain a life of leisure. Consequently, they never have to do all of the insane things they do — they do it because they love it.
Played with in Batman: Arkham City. Batman wasn't enough to get the prison city shut down, so he started putting on the pressure as Bruce Wayne. And was immediately kidnapped on live television and thrown inside.
Art of Fighting: RobertGarcia is the scion of one of the wealthiest families in the world and is the current CEO of the Garcia Foundation. He's also jointly responsible for taking down Mr. Big'ssyndicate and was personally responsible for laying the smackdown on Big, who happens to be one of South Town's most feared and powerful crimelords.
New Orleans oil-billionaire Spencer Troy is supposedly the President and CEO of Troy Petroleum. That said, to the general public, his "job" is fishing on the Gulf of Mexico, dating supermodels, and playing golf. At night he patrols the city as the heroic Battlecat.
Spencer Troy's daughter, Diedre Troy, is also a Rich Idiot With No Day Job. While secretly fighting crime alongside her father at night as Minx, during the day she does a remarkable imitation of the Paris Hilton/Lindsay Lohan party-girl lifestyle.
In Frisky Dingo, Xander Crews has a secret identity as Awesome X. He has no super powers other than his small army of mercenaries. This is mocked by others, as "your super power is management". Awesome X did kill every villain, but he is hardly heroic as he is a total dick. He's also not all that good at management.
In The Simpsons, Homer's family are apparently unable to convince him that Bruce Wayne is secretly Batman, because he's too much of a playboy to have the time. They are somewhat freaked out by his disturbing level of denial ("Why does he think Alfred's friends with Batman?").
Larryboy both plays this straight or averts this, depending on the series. In the standard VeggieTales episodes, he is never shown with a job. In the Cartoon Adventures of Larry-Boy spinoff, he's still a billionare, however, he works as a janitor at the Daily Bumble, the town's newspaper. According to Archie, his butler, this is so he can "keep his finger on the pulse of Bumblyburg". Why janitor? It was the only job he was qualified for. It's implied that Larry then has two secret identities, having to hide the Rich Idiot side from those who know him as a janitor.