Some characters eventually get to high enough levels that the game, and the GM, can't deal with them. Campaigns become too powerful, overbalancing, and players start to resort to munchkin-like behavior, sometimes just to survive. The characters, being such a high-level, are hard enough to kill as it is, but at this point (usually around 12th level if not sooner) can use so many means of escape or attack, that extremely powerful monsters or very large numbers of monsters start being treated like vain conquests and personal tallies.
Of course, it is possible to invent grossly powerful new monsters, but this leads to even more munchkinning, and who hasn't heard of a game somewhere where a party slew a tarrasque, or several tarrasques? The GM may feel like introducing deities to the characters and divine encounters on a regular basis become tedious and unremarkable, if not downright unbelievable.
There become too many possibilities for the average GM to control and the players and their opponents have so many choices open to them at such high levels, the common course of actio is just make monsters more powerful, with the result being that the characters defeat the monsters and become even more powerful. Soon we hear of 30th-level games where characters can turn into demigods themselves.
Luck becomes an even bigger factor in a game that already exists upon the premise of "whoever hits first will probably win"
The first solution is voluntary retirement. However, most GMs may be hesistent to force such a condition onto a player's character (I know I would be) and many players refuse outright to retire a character. When retirement does happen to a character, it's usually the player's idea.
The second solution is similar to the gunslinger stories or old western movies, the characters have become famous explorers and adventurers. Young swordsman and magicians who want a quick reputation will come looking for a fight, and often will cheat to win such a fight. If the characters operate as a group, then this will only occur on a larger scale, other gangs or travelling adventurers will try to put down the characters, or perhaps dog their heels in their next adventure, only to snatch the treasure away from them and flee at the very end (Belloch from Raiders of the Lost Ark comes to mind).
The third alternative is politics. Birthright tried to include this but failed miserably, succeeding only in creating Highlander-Fantasy. At higher levels it's possible the characters will become involved in larger issues that affect the world, wars, governments, etc. Their rewards are wealth and political power, but even in 3rd edition experience can be handed out for such "social battles" and these become similar power-mongering fights like described earlier, except instead of using Strength and a level-based to hit roll the characters are more than likely using their Intelligence or Wisdom and possibly a skill roll which has the potential for going overboard, "I have a +24 to my bluff skill so I'm going to convince the mayor that I'm from the regional government and he needs to give me his monthly tithe" However, if the characters are well-known they'll have to think of better ways of conning NPCs out of their cash then my feeble example.
Lastly, no one is going to rise to double-digit levels without making enemies. Even if they could, similar high-level characters will notice the PCs for the powers they can command. Someone of high-level who thinks the PCs might get in his way could possibly wish for a PC to become crippled by accident, just to get them incapacitated. This could start to get arbitrary, and if the PCs have to start casting divinations just to see who is adversely affecting them it could be annoying for the palyers.
In any system, characters will become powerful. How do you deal with it? Well, typically in D&D, you throw them into Hell. Just look at how many outer plane adventures are geared towards incredibly high-level characters. Yet, in many fantasy novels and stories where the hero must make a journey to some extraplanar or very far away place, the hero is almost always inexperienced and typically very young. Luke Skywalker (from Star Wars), Harry Potter, Bastion & Atreju (from the Neverending Story), Frodo & Sam (from Lord of the Rings), and Arthur Dent (from the Hitchhiker's Guide) all come to mind.
Something to think about, at least.