It very much depends on the DM.
There are lots of variations but there are two basic approaches.
First, what you might call the World of Warcraft approach whereas you level up the numbers get bigger but the fundamentally the game play doesn't change. You at high levels might fight Orcus and his court, but fundamentally that would be little different than fighting a goblin chieftain and his bodygaurds just with bigger numbers. It would be entirely up to the GM to try to invoke the flavor of epic scale through description.
The notion of the 1-20 "Adventure Path" is inherently tied to this idea with only small divergences when the writers provide hopefully engrossing minigames like the colony building episode in the middle of 'Savage Tide'. Fourth Edition embraced this model of high-level play wholeheartedly, with even 30th level being mechanically not that different from 3rd level.
The second approach is that as you level up, the focus of the campaign gradually shifts. As the players ability to influence events widen, they more and more become involved in shaping their destiny and the destiny of the world around them, and are less and less worried about tactical combat because tactical problems that would trouble them become rarer and rarer in their lives. The game slows down, even to the point of becoming dynastic - PC's start families, build empires of various sorts, and take their place on a political stage. Adventures of great import occasionally come along, but the sort of matters that previously occupied their attention are now beneath them and are delegated to lower level characters. Very high-level characters might enter semi-retirement, and players take up playing their main PCs retainers and henchmen. If the campaign goes long enough, the material plane becomes too small of a field of endeavor for such mighty characters, and they may begin to influence not just their world's politics but even their world's cosmology - becoming mighty figures of legend and song.
It's that later game that is implied by 1e AD&D, where anything beyond 10th level is considered "high level".
As for my own experience, meeting regularly it might take 3 to 6 months to gain a level. Even with new PC's introduced at close to the highest level of PC in play, it might take years to reach high level play. My recent seven yearlong campaign took a hiatus at 10th level. I would imagine very few people reach super high levels without deliberately rushing through levels simply to reach the higher levels as if leveling up was the goal of play, and if they do that and then whine about how high-level play isn't that fun, well I don't have much sympathy for them.
The biggest difficulty with high level play is figuring out how to balance it so that combat or any other challenge is meaningful. There are some inherent limitations imposed by any fortune system once characters are skillful at something in that the range of skill between party members diverges over time. At low levels a character might be more skilled at something than another character, but every character can be challenged to some degree by the same problem. At high levels, whatever would challenge the skilled member of the party is basically impossible for anyone else in the party. This means it's hard to have situations where everyone is contributing and challenged without risking squishing some member of the party on a single bad roll or unfavorable fictional positioning.