There are several answers, but all of them are problematic.
Stacking bonuses to dispel checks
There are various things that add a bonus to your dispel checks such as the Inquisition domain (Complete Divine), the master abjurer prestige class (Complete Mage), the spell spellcaster’s bane (Complete Mage), or a dispelling cord (Magic Item Compendium). Is this allowed to go over the +10 or +20 limit (or the +25 limit of reaving dispel from Complete Arcane or chain dispel from Player’s Handbook II)? No one knows! But maybe.
Anyway, there are a finite number of these, so eventually you’ll run out and won’t be able to pump your dispel check any higher, so this only delays the question from ~30th to maybe ~50th or so. Even that might be generous.
You’re an epic character, so you have 9th-level spells, and Mordenkainen’s disjunction is one of those. It doesn’t even bother with a dispel check, so you don’t have to worry about it.
There are a few problems here: Mordenkainen’s disjunction is seen as the nuclear option for a reason. Most players don’t want to use it because they don’t want the DM using it, because it is utterly brutal and there is absolutely no defense. If Mordenkainen’s disjunction becomes the norm, mages will simply refuse to be in one another’s presence for any reason; all communication can happen through spells, and combat can happen at long distances, or, if absolutely necessary, through scry ‘n’ die tactics. Under no circumstances will it be acceptable for another spellcaster to cast any spell in your presence. That... changes the game. A lot.
The other issue is the thing with artifacts: if your target area includes one, you have a good chance of becoming an epic commoner. That is a career-ending move for a mage—for the player, it’s time to roll up a new character. Disjunction emphasizes how there is no coming back from this. That’s bad.
There’s a Dispel seed for creating epic dispelling effects. The dispel check starts as 1d20+10, but you can increase the +10 by +1 for each +1 you add to the Spellcraft DC to create the spell.
Epic spellcasters ought to be able to hit patently absurd Spellcraft DCs, far in excess of what a caster of the same level could hit using their caster level. That makes this as good as disjunction if you put some effort in, without the risk of commonerization. As you level, you may have to research new versions to improve the check, but that’s hardly a big deal. It can also affect artifacts, but presumably—the rules don’t actually specify beyond saying it can affect them—it just suppresses their magic temporarily, much as dispel magic does on magic items. Certainly there is no suggestion of losing all spellcasting.
There’s also the pre-statted epic spell superb dispelling, which has a dispel check of 1d20+CL, max 40—which is weird, because that’s not how the Dispel seed works. Anyway, you can do far, far better than that abusing mitigation.
Reserves of Strength, at least if able to twist RAW
Arguably, Reserves of Strength does this, RAW, though it’s almost-certainly not supposed to (and even the RAW argument is extremely contentious).
When you cast a spell, you can decide to increase your caster level with that spell by 1, 2, or 3, but you are stunned for an equal number of rounds immediately after doing so. […] You can exceed the normal level-fixed limits of a spell with this feat,
(Dragonlance Campaign Setting)
The problem here is what “with this feat” means—does it mean “with the bonus granted by this feat,” or does it mean “when you are using this feat,” or even “when you have this feat, whether you use its other effect or not.” As I said, it’s probably supposed to be the first one, and the other two really require putting blinders on and willfully choosing not to consider the context, but the argument is made (I’ve made it myself in other answers) that when you have this feat, or at least when you use it, it removes the caster level limits on the affected spell entirely, not just for the 1-3 bonus caster levels. So you can have CL 40th, and cast dispel magic, and use Reserves of Strength to add +1 CL to it, and then you get to add +41 to your dispel check because it removes the cap entirely. Realistically, they probably just meant you could add +11 in that situation.
Reality: Epic just isn’t balanced
That paragraph I wrote about how Mordenkainen’s disjunction warps the game is kind of disingenuous—optimal spellcasters are already behaving like that before they even get 9th-level spells, to say nothing of epic spellcasting. Scry ‘n’ die should arguably be the norm once greater teleport is available. Plus actual invincibility is available through a wide variety of means by the mid-teens. You can avoid these things, but it takes conscious effort not to “god mode” out of the game for some spellcasters before even reaching 20th.
The problem with this is, Epic Level Handbook posits that you can continue playing the game as characters get even stronger than that. Epic spellcasting is completely broken, and even if you ban that, just taking Improved Metamagic and Improved Spell Capacity can make even those spellcasters who didn’t abuse metamagic even more broken than an incantatrix.
So there’s no reason why a character at that level of power should even be playing this game: why risk anything by going toe-to-toe with a dangerous foe, removing their defenses while you allow them to do the same to yours, when you could just... not? Cast the epic spell “solve my problem,” and be done with it. Or abuse love’s pain, or perform the ultimate scry ‘n’ die, or transcend mortality entirely and become a deity. All of these are things that are supposed to be easy for a spellcaster of this level. But if you can do all of those things, there’s nothing left to play for. The only reason the world of D&D continues to function despite the fact that epic characters are statted for it, is because none of those epic characters actually use their abilities to anything like their potential.
Which offers an out, of sorts: you can just ignore all these abilities that ruin the game. The problem with that is, when you could do anything, and you have to just arbitrarily limit yourself, then the rules aren’t actually helping you create the game you want to play. The rules tell you that you can do all of these things; it’s you who is deciding not to. Which means that, at that point, you have to make all the judgment calls yourself, which means you’re playing freeform. Which is a fine thing to do, people very much enjoy freeform roleplaying, but at that point, why waste time consulting D&D rules that you’re going to have to wind up ignoring anyway?