So my question to you guys is, what sorts of problems do you see happening a lot at Epic Levels, be they conceptual, mechanical, or otherwise. As a bonus, feel free to give a pointer or an idea on how to remedy that problem. It doesn't have to be good, but it could start something somewhere somehow.
Having played a significant amount of epic, I think there are three take-home points. First, there are some basic system assumptions that begin to fall apart at high levels. Second, that trying to play a high-level game the same way you played at low levels probably won't work. Third, that if you're willing to adapt to a slightly different playstyle, it can be really fun. I learned a lot about D&D by playing and DMing at epic levels.
To get the ball rolling, I can see that once the modifiers get staggeringly high, the 1d20 dice that the whole game is centered around becomes dwarfed by the sheer size of the modifiers that come up.
This is the perfect example to me of something that can be either a problem or opportunity. The way d20 math works, numbers scale really fast, and epic characters are inconceivably good at what they do. But if you want superheroes, this is okay. Particularly for checks that don't fail on a 1. If your bonus in a skill is X, you can do any task at DCs up to X+1 automatically. Taking 10 and 20 extend this even farther. Thus, epic play is a great opportunity to abandon the constant d20 rolling for every little task, and simply have characters state what they want to do and have the DM adjudicate it. Epic play can be fast and rules lite if you approach it with this mindset.
Then again, the relevance of the d20 is still worth addressing. I use the scaling d20 variant (where, for attacks and saves a 20/1 is not an automatic success/failure, but instead you add 20 or subtract 20 from your result and roll again).
Epic does create epic bookkeeping problems, which are hard to really resolve, but it does encourage players to play bookkeeping-lite characters and encourages DMs to be fast and loose with tracking resources, both of which I think can be good things.
There's a big issue in designing appropriate challenges. The epic NWN expansion just sends you through dungeons with epic warriors swarming you. That's stupid. In an epic game, combat needs to be rare and have great significance, and stakes.
The epic spell system is a cool idea; arguably closer to the way magic should work than the actual magic system. But it's so raw and so abusable that it requires quite a bit of thought and interpretation to work. Both the DM and the players have to trust each others discretion.
A big system issue to me is that epic characters get too good at things they should suck at. Your epic wizards can be pretty good with melee weapons. Gandalf in the LotR movies notwithstanding, wizards should never get a +20 attack bonus (barring various exceptions like Tenser's Transformation and multiclassing, obviously). That one's hard to fix.
Another problem is simply that power corrupts. PCs can become total psychopaths if they think nothing can hurt them, but simply having watchful NPCs restrict them is not a viable solution in perpetuity. Frankly, as a DM you just have to learn to trust your players to have some sense.
