Money makes you lonely, foolish, or even evil, but worse than any of these: since you can buy anything, including having people do everything that needs to be done, you are bored, bored, bored. There are no more challenges in life. You don't have to work. You don't have to keep house. You don't, in fact, have todoanything.
And everyone expects you to like it.
Even the Royal Brat, suffocated with toys, may suffer from this and straighten out when given something worth doing. Many rich characters find themselves abruptly precipitated into adventure and (once they get over the shock) loving it because it lends meaning to their life.
More proactively, the Rich Idiot with No Day Job, Gentleman Adventurer, and Gentleman Thief all often turn to In Harm's Way to escape. Other characters may turn to charity works, travel, writing books, scholarly work, or other ways of becoming Non-Idle Rich to avoid this problem. The Upper-Class Wit may be warding it off with his observations. Expect it to be a feature of any Gilded Cage.
Not all the Idle Rich are bored; the intelligent, the well-intentioned, and those with a lot of energy may be alone, and the family may regard their boredom as silly, or recommend shopping as an infallible antidote. Those who like idleness will often clash with Non-Idle Rich in their own family.
The courtiers of the Deadly Decadent Court are uncommonly likely to suffer, despite their intrigues and their culture.
Compare Victory Is Boring; overlap is possible. Can justify Cut Lex Luthor a Check; they have enough money so the point of their Evil Plan is excitement
Love And Rockets - Young Luba is bored with her life as a rich housewife, and goes looking for excitement. She finds it in drugs, affairs and dancing
The British comic Whoopee! featured the Bumpkin Billionaires, a family of yokels who had come into some money but were desperate to get rid of it and go back to their old life. Their attempts to rid themselves of their wealth was always unsuccessful and often ended up increasing their wealth.
The British comic Lion had the anti-hero The Spider (not the same as the character created by R.T.M. Scott) whose back story is that he was a bored, rich, super-fit middle-aged man who initially took up cat burglary but then decided it was too easy, returned everything he'd stolen and became a crime-fighter instead.
Benito Medici, the son of one of the richest and most powerful men in the world in 100 Bullets. He's so bored that he gambles just for fun and gets involved with some very dangerous people out of his father's reach, just for fun.
Defied in one Dilbert comic strip. Dogbert, due to some very good investments, gets a few hundred thousand from said investments. He's out for a walk, when he tells a random businessman "Roll around in that mud puddle over there and I will give you $1,000 dollars." The businessman does so, with Dogbert quipping "I don't know how rich people ever get bored."
Rat Race: The film is about a group of rich people who shove two million dollars into a locker several hundred miles away, tell a few normal people where it is, and then take bets on who will get there first.
As well as bet on random things during the race.
Surviving The Game: The rich hunting clients all pay $50,000 in order to go hunting - hunting a human, that is - for the ultimate rush.
In P. G. Wodehouse's Summer Moonshine, Mr. Bulpitt has only once failed to serve processes on anyone: a multi-millionaire who died leaving a note that Mr. Bulpiit had cured him of "onwee" as Mr. Bulpitt says it. Also, he had left him his entire fortune for it. Mr. Bulpitt retired, briefly, and found out what "onwee" was, but even back in business, he can save his Impoverished Patrician niece.
In "The Case of the Rich Woman", one of Agatha Christie's Parker Pyne stories, this is the problem that the rich woman wants Mr. Parker Pyne to solve for her.
In Samuel Johnson's Rasselas, Rasselas's problem in the valley is exactly this, because he is surrounded by every luxury.
In John Buchan's The 39 Steps, the narrator Richard Hannay Jumped at the Call because of this trope. Later books find him happily working hard as a regimental officer in the ensuing war.
In Dorothy L. Sayers's Gaudy Night, Harriet indignantly defends Lord Peter Wimsey: catching murderers, even for fun, is difficult and dangerous, and many people have reason to be grateful for it.
Jed Clampett of the Beverly Hillbillies. He has millions due to selling his land to an oil company, but is many times shown being bored as he would rather work than be idle rich. He enjoys doing work like mending, repairing, or gardening, but of course Mr Drysdale won't let him work a job because it's embarrassing to him. There's also no place for him to do much hunting or fishing in Beverly Hills.
On The Twilight Zone in the episode "A Nice Place To Visit": A small time crook is killed and ends up in the afterlife where he gets whatever he wants. It soon gets incredibly dull and the crook asks why he wasn't sent to "the other place" (i.e. hell). It turns out this is actually "the other place" and getting what he wants all the time is his punishment.
An episode of CSI: Miami has rich people pay to hunt humans.
Paper Mario: The Thousand-Year Door: Flavio, self-proclaimed "Trader extraordinaire, millionaire sailor of the seven seas" sets out on a voyage with Mario just for the thrills, as he's grown bored of hanging out in the tavern staring at his jewel.
SpongeBob SquarePants: Mr. Krabs sells the Krusty Krab for a lot of loot. It's not long before he runs out of things to do in his retirement, goes on a rampage that destroys the Krusty Krab, buys the destroyed restaurant back, and it's all fixed up the next time we see it (in the next episode).
Before the rampage, Mr. Krabs became a dishwasher at the restaurant just so he'd not stay bored. If he didn't care about how Mr. Blandy was running the place, he'd gladly remain a dishwasher and treat his former employees as equals. He even allowed Spongebob to call him "Eugene".
The Batman: The Animated Series incarnations of the Terrible Trio are young millionaire scions who turned to villainy for no reason other than boredom.
This is the reason for the Guild of Calamitous Intent on The Venture Bros. To keep the idle rich occupied so they don't hurt other people.
In King of the Hill when Mihn and Kahn finally manage to join the prestigious Nine Rivers Country Club this ends up being the case. Mihn is openly bored during a discussion of the values of pairing different cheeses and wines while Kahn only feigns interest (and comprehension) to impress Ted Wassonasong.