So we all know the general issue with these sorts of campaigns.  Your party is so strong, that basically nothing you throw at them will reasonably pose a challenge.  So the question is, can you think of any good story lines where even being demi god levels of powerful still means basically nothing in the face of the next threat they will face?

I'll go first.  The party has to somehow sabotage a diplomatic meeting between the Archdemons and Archdevils.  If they fail, the two forces of hell will combine into a single unified force that will lay waste to the various realms of existence. 

Over the 20 levels of play the DM will have gotten a good idea as to what motivates the PCs. Thus, rather than throwing powerful monsters at the PCs which they can easily avoid, provide them with an RP situation where they have to potentially sacrifice themselves (for the greater good).

If it's a campaign idea discussion, that heavily depends on your players and how they've gotten there; like Farling said.

For a one-shot, just have them do PvP or something. They have no advancement and no real threats to the group as a whole, so might as well make it a tournament thing to become the New God of the setting.

God Slayers. All gods have turned evil and Eldritch, and you must kill them. A few fun boss ideas were spawned, then I left the concept alone.

The idea of a Level 20 Campaign is the myth.  This is one of those places where I find Pathfinder has a leg up on D&D.  There are a lot more creatures.  You actually have information for beyond level 20 and you also have the Mythic series for creating god-type heroes in the making.  I've never actually seen a campaign make it from a lowly status all the way to level 20... and I'm not going to start any campaigns that begin at Level 20.

But if you did... I would start with Dragons and work into other planes very fast (and back and forth).  The 'go fetch' quests would be some kind of ritual object needed to weaken an ancient dragon who had the name of a pit fiend that must be slain to reveal the lynch pin of an organization that has deific backing from Mount Celestus in the Feywild... etc.  It basically is Monster Hunter, but with some kind of chain progression.  Or else you are a Dragon Slayer club and the notoriety and privilege are tied to rank which is tied to the might and power of dragons you have slain.

By my logic, the best challenge for a level 20 party would be a criminal mastermind of epic proportions.

So, perhaps there is a BBEG who is working to bring about an apocalypse or incite a war between the material plane and the plane of fire, or to collapse the planes into one plane of utter chaos, or to summon an old-one style god to enslave or destroy the world. Big stuff.

Then, you need to separate them from the PCs. Perhaps they are asked to enter a dungeon to recover an artefact which the Great Seer has said must be recovered, and they enter this demiplane which is a big challenge for them (but not too big because, you know, level 20), and they find the item is gone.

The game then progresses with the party trying to recover this item, or others, to find this mastermind. They might have to work out the pattern to get ahead of them, or to work out the final goal to try and counter it. The game should play as a puzzle, where each step, once revealed or worked out, can be tackled using all the skills they have learnt by being level 20 - it should be stuff they can deal with, because the issue is not whether or not they can defeat the dragon and find the arkenstone in it's horde, it's how quickly they can do it. Perhaps the dragon has the arkenstone on its chest, and it flees after the party launches their opening assault, and perhaps that is what allows the enemy agents to retrieve it first.

Make things increasingly frustrating that their strength and powers are no match for the plans of their enemy, and when the ritual is started - as it doubtless will be, as their faceless BBEG will not be revealed to them, so can just keep trying - then they might find the reverse -their powers are nothing against an elder god, and they have to instead tackle the ritual sites to stop the ritual. They might even have to split the party to make it happen in time. 

In short, the level 20 people can fight anything - except time limits and unseen forces in the background. If you give them things to punch, they will win. Go Lex Luthor on them instead.

Have a BBEG who is building an army--creating clones / simulacra of the level 20 players.

Or maybe he's from another parallel timeline / dimension where they are evil.  Or perhaps there is no guy, it's just their PCs from another timeline/dimension/plane.

The BBEG creates these clones based on the party's characters.  Now, they are level 20 characters going up against their level 20 doppelgangers.

Maybe the only way to defeat these bizzarro-world doppleganger baddies is through the power of teamwork, or (preferably) something else that's more clever.  Maybe they have a gained the favor or friendship of a powerful NPC who can help, or perhaps the magic items they've chosen along with the plan they hatch is the difference between failure and success, life and death.  

Maybe they TPK early on as lower (10-15th) level characters as part of the story (session 1), and much later on those characters return, controlled by a necromancer who has leveled them up to 20 in a unique way.  Maybe they're back now, improved with artificer infusions, or they have been subjected to experiments and now have mutant limbs/powers, or they now have large companions they ride on that burrow/fly, or perhaps they've been taken over by illithid tadpoles.

The party needs to bring an artifact back to the king to save the kingdom.  Conveniently they are transported to the room holding said artifact to begin.  They then discover that they are many levels below ground and must make their way to the exit.  No big deal for say, 4-5 level 20 characters?  Teleports and any other magical travel spells somehow drop them back to the bottom level.

Hmm, now as they make their way out, it seems that about every 10 minutes, 4-6 CR 15-18 monsters appear and attack them.

With a "dungeon" 6-8 levels deep, and some labyrinth type navigating, you will wear down these god like heroes through simple numbers.  No time to properly rest and recover for the most part, meaning resource management will be critical.  Add to that, some resources likely pissed to the wind as they TRY to teleport/shift/magic themselves out.  If the encounters STILL seem to trivial and not spending resources to handle, raise the CR or number of foes that appear.