I agree with a LOT of what Celebrim says, except the part about it ending at 12th level.
I played numerous characters above 20th level, including a 33rd level cleric, and never lacked for content or challenge.
This almost requires a fork.
While I know of campaigns that went on for a long time at high levels of play, its also true that a party of 33rd level characters is not challenged by anything in the DMG and generally isn't challenged by most things in the Dieties and Demigods either (except dieties with absolute non-stated abilities).
The Monster Manuals feature monsters rated in difficulty as I, II, III, IV, V, VI, VII, VIII, IX, and X. Level 'X' monsters, or level 10, were the hardest challenges in the official canon, and they roughly correspond to things found on the 10th level of the dungeon and which challenge 10th level characters. Realisticly, the 'X' category is fairly broad, as Orcus or Asmodeus is a bit harder challenge than say a 14 headed pyro-hydra, but even if we extend out this category realistically we soon find that by about 12th level nothing in the Monster Manuals represents a really serious challenge to a prepared party. This goes double if the fights are done as straight up fights in 30'x40' empty rooms. By arbitrarily increasing the number appearing of level X monsters, equiping them with the best and most lethal items, and by placing them in hideously lethal environments, we might be able to extend play out to 15th level or so. But by that point, we are already dealing with two problems. First, if we give the best and most lethal items to the foes, then they will soon be in the hands of the PC's, which makes challenging the PC's even harder. Secondly, by increasing the numbers of foes, we risking making the play more repetitive, slower, and less interesting. For example, above some number N, a monster ceases to meaningfully exist as an individual and becomes a mere terrain problem for a high level player.
Consider these situations:
"You see probably 100 rust monsters."
Me: Err... the ceiling is 80' here? Reverse Gravity. Dismiss Reverse gravity. I believe that should solve the problem. If it doesn't, we'll just fly through above their reach on the magic carpet.
"You see the passage filled with thousands of carrion crawlers."
Me: Sigh...I spend 20 charges from my wand of fire. With a volumn of 30,000+ cubic feet per fireball, that should pretty much clear the entire chamber.
And the same basic issues apply to scores or hundreds of more powerful monsters, and even were it doesn't it just gets boring to play out the inevitable. The monster to hit table maxes out at 16 HD, and therefore if you have a -10 AC, pretty much no monster in the game hits regularly you without signficant aid. Conversely, few monsters in the game have an AC of better than -3 and between the advantages of having high strength, magic weapons, and a more open ended attack table, at higher levels a PC fighter basically can't miss and does more damage than pretty much anything in the game short of a greater deity. And that's to say nothing of the open ended fireball damage or the numerous ways a creative or even uncreative spell caster can just say, "Die no save." Likewise, with the advantage of attributes and magic equipment, high level PC's have better saves than anything short of a demigod. Likewise, because PC's are eligible for a constitution bonus and monsters generally are not, if they have one, they'll quickly end up with more hit points than anything short of an arch fiend. It won't be unusual for the fighter to have more hit points than a dragon (and in fact, its pretty essential) or really anything else in the book. The list of monsters in the book capable of challenging the party at higher and higher levels rapidly diminishes to a handful. With a few extra moments thought I could list the whole list (about 2/3rds of them would be monsters that can also be high level spellcasters).
One possibility this pushes you toward could be called the Gygaxian solution, that is, if the problem is too much player power - take it away. Declare that no spell above 3rd level works, that all metal armor corrodes away, that spellbooks are erased absolutely, that many key spells cease to function, that saving throws are not allowed, and so forth. But like increasing the numbers of monsters, a little of this goes a long way.
Another possibility this pushes you toward is solo play. While even arch-fiends are pretty much pushovers for a 15th level party, they are much more challenging if you have to face them and their minions solo. However, again, this is mostly fun only if you just have a DM and a single PC.
Another possibility this pushes you toward is DM's cheating. Some of the high level games (ok, all of them) I was familiar with remained 'challenging' solely because the DM's pet reoccuring villains (whether it be a pet race like Drow or a pet character like a Vampire spellcaster) acquired at game time whatever abilities and hit points where necessary for them to survive whatever the PC's did, and always remained one step ahead of the players by the simple expedient of the DM devising every counter on the fly as soon as the PC's declared their intentions. That works, but needless to say gets old quickly (unless its the only game going, in which case, what the heck).
Now, that isn't to say that it can't be done. Alot of the innovations that 3e adopted that allowed for more open ended campaigning are things that you could do in 1e - give monsters attribute and therefore the associated bonuses, double or triple base monster hit points, increase dragon hit point/HD, give monsters class levels, make crap up, etc. Indeed, some people (including me) had started doing some or all of those things by the time 2e rolled out. However, from the 1e books themselves, its not at all obvious that you should or even can do those things.
