Honestly the system was showing its strain even at around 9th, where the mage spells (especially) start getting Munchkin, and also where the rest of such a party has major loot including (in AD&D) artifacts. And, you know... what's next for a player who's bored with bashing the same sort of monsters just at higher levels?
Frank Mentzer staunched the bleeding with the Companion Set rules... somewhat. Ruling dominions and grand-strategy, hey, we'll take it; and we liked it. But even he had to come up with a Master Set, which he sketched out as the path to Immortality. Sorry, lads; the M- series of modules were (with one or two exceptions) ass. All three of the I- series were grainy polypey rectum.
Eventually 3e players wanted a Master Set of their own so Wizards printed the Epic Level Handbook for them - complete with apotheosis! Problem: it was retardedly broken.
The Handbook hated martials even more than normal 3E. While epic level Wizards got feats that could quicken every spell for free, cast more spells than 1+1 quickened, or use "epic spellcasting" (more on that mess in a bit), a Fighter got feats that gave him +4 damage if the target was within 30 feet. Fighters didn't even get higher attack bonus than anyone else when advancing past 20!
Perhaps the biggest mess is epic spells. "Despite their power, epic spells still follow the basic rules for casting spells, except as specifically noted otherwise.", with the "noted otherwise" being pretty much everything except how Dispel Magic works on them. Unlike normal spells, epic spells are all created by the player (though there's a handful of really terrible examples). Not the player character, the player. Epic spell are essentially a big custom spell system that you need to meet a skill DC (very easy to munchkin to a far higher level than intended) to make and use. Unlike most "custom ___" systems, these aren't suggestions or guidelines (except in the way all rules are), so a caster can do anything they can get a high enough skill check for.
If you wanted even more broken shit, the Draconomicon states that "Dragons of at least old age also can choose [epic] feats even if they have no class levels" and "true dragons are those creatures that become more powerful as they grow older". The Dragonwrought feat (which can only be taken at level 1) for Kobold turn them into dragon type creatures that don't take penalties (only bonuses) for aging and gives them age categories. This means you don't even need to be epic level to take the bullshit epic feats.
The one thing that people actually use out of these rules are the skill uses, in large part because they aren't actually tied to epic levels and can be made by normal characters (albeit with some focus required). With very high ability in a skill a character can create truly impressive effects. The most infamous uses are the Diplomancer, able to make any enemy a fanatic ally in seconds, and Escape Artist allowing you to go through spaces smaller than your head, potentially even into other people. There's even uses for skills that should have been in the core book like hiding someone or avoiding suspicion when asking for information.
4E avoided segregating epic rules and instead used Epic Destinies to control high level progression. It's probably the best ruleset for "epic levels" any edition's ever had.
When the wheel of time turned and brought us 5th Edition D&D, the entire Epic system was replaced. Gone are Epic Levels and Epic Spells. In their place is the new Epic Boon system. When a character hits level 20, they reach something called the Level Capstone, which prevents them from levelling further. However, they continue to gain XP. When they reach 405,000 XP, they can gain an Epic Boon, to reflect how they have slowly started to become one with the Weave of magic that keeps reality in line in the D&D multiverse. There is no limit to them, and you get another every 50,000 XP after the first one. These Boons are quite varied. Some are junk, like getting a Feat (which some classes have had nine chances to do already), while others are amazing, like getting six more spell slots or becoming immune to hunger. Some are just weird, and allow the character to do something their class would normally never get to do, like a Bard getting a Paladin's immunity to disease.
Pathfinder went another, though equally broken (both in the "overly powerful" and "this shit doesn't work" sense) idea with Mythic Adventures. Rather than increase level directly, a character got additional "mythic tiers" that gave him new abilities. A character could be level 1 with mythic rank 10 or (more likely) level 10 with 2 mythic ranks. Like Epic Level Handbook these abilities varied between stupidly OP and weak even for normal options. Also like ELH, most of the stupidly OP stuff went to casters and the weak stuff went to martials: Casters get to cast any spell with a standard action casting time on their list, regardless of if they know it, while martials can move and attack at the same time. Universal options tend to be better than the ELH's marginal improvements on normal options though, having stuff like becoming a demigod that can grant spells. Classes vary wildly on how well they work with mythic abilities: While your typical full caster and beatstick are compatible enough, anything slightly unusual is completely and utterly unsupported or so poorly supported it might as well not be. Classes with "pets", such as a Druid's animal companion or Summoner's Eidolon, are boned because mythic doesn't advance their pet (and the Mythic Companion ability is very lacking), a caster with medium BAB and 6th level spells (Alchemist, Bard, Magus, Warpriest ect.) has to choose from focusing on being a melee killbot or trying to ape the full caster's trick with even less support while being levels behind them while having absolutely no support for their unique abilities.
In place of epic spells is alliterative Mythic Magic. By spending Mythic Power on a spell you can break limits on it and do new things. Unfortunately the system is gimped by how, instead of being something anyone with mythic power and casting can do, it requires the caster take a feat/path ability that gives access to a single mythic spell per mythic tier, meaning pretty much all of the interesting ones will never, ever be used since the opportunity cost is, literally, apotheosis. There's the Ascendant Spell metamagic that allows converting spells into their mythic version, but at a +5 spell level increase and a large chunk of spells bared from using it, it is almost never worth it.
Paizo pretty much dropped it when they realized how shit it was, but it did survive for one use: While the system is horrible to give your PCs, it's really good for making boss monsters, especially so if the GM is reluctant to make stuff up from scratch. While giving players extra actions, immunities and powerful new abilities is a good way to destroy your game, boss monsters enjoy being able to act multiple times against a group of multiple PCs, not getting save-or-died, and having some flashy abilities random encounters won't have. Even Paizo acknowledged this, as within a year the only time they ever referenced the book for was some boss monsters. Mythic eventually receive new material in one of the system's last books, but it was primarily to add abilities compatible with newer (mostly occult) classes.
Paizo's screw-ups didn't stop third party companies from taking a crack at the system, however, and in Pathfinder's usual trend these companies actually improved upon the system a great deal. Legendary Games' Mythic Heroes' Handbook offered revised rules for Mythic that nerfed the absurd rocket tag of the system into viability. They also provided options for many of the classes that Paizo left behind and re-organized many features so that classes were no longer locked into a specific path. This was successful enough that they put out several more books in this line and Legendary Planet, a full mythic adventure path.
The Wrath of the Righteous video game also has its own take on the Mythic Path system.