My experience with high level 1e AD&D is that the big issues tended to start to become problems which couldn't be solved merely by killing something. You sort of end up in the same problem as Superman - you may have unlimited ability to smash things but if the problems faced by humanity can't always be solved just by smashing things.
Our campaigns had a tendency to become political and our problems started having more to do with economics, persuasion, and military force than only going into a dungeon and killing something. We started having as many foils as foes - people that opposed you but weren't really bad guys. You end up in situations where killing the obstacle makes the situation worse, because you then look like the bad guy (and interestingly, that perception might even be correct!).
We also had a tendency to go solo more often. Many play sessions were one PC + the other players taking the role of that PC's retainers (themselves quite high level individuals). You might have one 12th level PC and 4-5 7th-9th level psuedo-PC's (technically or originally NPCs but now often as not ran by a player).
For the main PC's obtaining power had less to do with getting bigger fatter loot, and more to do with attracting more and more high level followers (or preserving the low level ones until they leveled up and could handle themselves), and expanding your political sphere of influence. Even leveling up was relatively less important, because it took so much XP to gain one more level that unless the DM deliberately set out to level you up (for example, see the GDQ series) it wasn't going to happen anytime soon. Becoming persnally more powerful didn't necessarily help anyway. Sure, you might well be fighting off invasions of tentacled things from the outer dark, but the scale of the problem was such that you simply couldn't be everywhere at once to handle it. It doesn't help that you can kill 300 orcs or equivalent by yourself, if the other 29700 overrun the city and slay its inhabitants while you are doing so. It doesn't count as a great victory if you save this city if it means four other ones are burned to the ground. You can kill this giant or that dragon, but when the problem is on the scale of hundreds of same an army became essential if your goal was actually 'saving the world' or at least your corner of it.
The idea of going one on one against the bad guy seemed to fall away by 10th level. The big bad guy was another Lord like you with minions and an army and he wasn't about to just go toe to toe with a bunch of high level characters if he could avoid it, which he usually could.
For me, one of the things RPGs seem to be getting away from that isn't entirely 'win' is subsystems. Sure, subsystems can be klunky and inelegant, but nothing beats a subsystem for having flavorful, quirky, and surprising mechanical results. First edition D&D was the king of subsystems. Things can happen when you have subsystems that are pretty much impossible when everything has the same neat tidy sort of mechanical resolution. Part of what made high level play different from low level play was access to new subsystems. That, and because the systems weren't linear, they had emergent properties as you progressed along them. So long as you have a rules set with a single generic system, its very hard to get away from the feeling that the game never changes. This is especially true if the core system by design plays out the same way at high level that it does at low level.