Can you increase an ability score above 30?

At the beginning of most source books, there is a rule stating that the rules are basically loose guidelines, and everything in the books are up to DM discretion. This is something to talk to your DM about, or if your the DM come up with your own standpoint. Personally I'd leave it up to the context of the campaign at hand. Ask questions like "would allowing this player to raise their score any higher cause the game to derail or lose it's meaning? How could this benefit the table? What are some potential drawbacks?" Decide for yourself because you'll never get a clear answer from a community split between rules lawyers and rule of cool fanatics.

Quote from TheGnome5786 >>

Whats to stop an Elf who might live for 1000 years from looking and finding immortality? Or at least something that lengthens his life two, three or even more times?

He has all that time to work out the magic, to strive for and possibly reaching demi god status.

Of does he just wait for death for a thousand years?

 This is not a PC, this is an NPC. Probably a BBEG.

You guys are having this discussion as if its in any way reasonable for a campaign to span centuries for the gratification of a single character. That's just not how D&D works, unless you specifically want it to work that way. The DM is in charge of the campaign, they don't just helplessly sit back as a player declares they spend a couple millennia getting 50 STR.

Quote from scatterbraind >>

 This is not a PC, this is an NPC. Probably a BBEG.

You guys are having this discussion as if its in any way reasonable for a campaign to span centuries for the gratification of a single character. That's just not how D&D works, unless you specifically want it to work that way. The DM is in charge of the campaign, they don't just helplessly sit back as a player declares they spend a couple millennia getting 50 STR.

 In a world full of NPC's thats sort of the point. Where are they?

What do Elves do all their adult life? Do NPC's just sit around all day in your worlds waiting for some PC to come by and give a purpose to exist?

Quote from TheGnome5786 >>

Its called magic.

 

The original poster was asking about a magic item that raised your charisma over 20 then shut off for one hundred years. It did not go away just shut off. If he lived long enough every one hundred years he could use it again.
There were no hard limits on the magic item.

 I would say that 30 is a hard cap no matter the magic. The physical (mortal or immortal) form just cannot go any further. 

just like no matter how much magic you can magic you will never cast Meteor Swarm beyond 9th level. You will never get a 30d6 fire, 30d6 bludgeoning or higher. 

Edit: and maybe the gods just won’t let them go beyond 30. 

From a rules perspective, 20 is the  cap for adventurers (see https://www.dndbeyond.com/sources/basic-rules/using-ability-scores#AbilityScoresandModifiers) and 30 the cap for monsters/etc.  The language says "as high as" which is a hard cap, not a soft cap, which means only specific effects would override it.

Certain items and effects allow you to exceed 20 as an adventurer via their specific rules, but technically, as a humanoid, adventurers are also monsters, which also have the 30 cap (see the definition of humanoid here https://www.dndbeyond.com/sources/basic-rules/monsters#Type as well as the rules for each race that declare your humanoid (or other) status).

Taking the rules into effect then, an adventurer who is allowed to break the 20 limit would still be subject to the 30 limit, unless an effect allows this to be broken as well (which no published rules allow)

From a gameplay perspective, the limits and modifiers are set based on bounded accuracy. the limits of bounded accuracy ensure game balance, and having PCs with scores exceeding 30 would essentially trivialize all encounters.

Characters have Legendary Actions

Quote from Sigred >>

20 is the normal cap for PCs. That can be raised via things that explicitly raise the cap (such as these books, boons, Barbarian's level 20 feature, etc), so I wouldn't place a ton of stock in 30 being a hard-cap for the entire system.

That said...

Quote from Monkey5210 >>

Out of curiosity, with the boon of immortality and something like the Tome Of Leadership And Influence, if you spent enough time hiding in a hole, could you potentially increase your charisma to say- 40? What's your guy's thoughts on the matter?

 

 ... no, you definitely can't do this. You can only benefit from reading the Tome of Leadership and Influence once. Even if you're alive long enough to encounter the book more than once, it's still the exact same book you've already read and benefited from.

You would have to wait a century to benefit from THAT Tome of Leadership and Influence again. But what if you find another one? The Tome is Very Rare, not Unique. 

I know 30 is max, how would you even raise an ability to a 30?  Obviously magic and books, but can you still get that high?  Specifically, how would you do the below?

Because of ASI level boosts your DEX is now at 20 and you are level 12.   How in the next 8 levels do you get a Dex to 30?  You can't use ASI anymore.  

This would be a specific example so no generalities and have to be in 2024 books, nothing in 2014 rules or books would be acceptable.

Quote from Character77006 >>

I know 30 is max, how would you even raise an ability to a 30?  Obviously magic and books, but can you still get that high?  Specifically, how would you do the below?

Because of ASI level boosts your DEX is now at 20 and you are level 12.   How in the next 8 levels do you get a Dex to 30?  You can't use ASI anymore.  

This would be a specific example so no generalities and have to be in 2024 books, nothing in 2014 rules or books would be acceptable.

As was mentioned above, it's possible to use more than one of those Tome/Manual of Whatever items. The 2024 rules also have Epic Boon feats, and the DM is always empowered to grant additional increases (e.g. via the Supernatural Gifts options in the Dungeon Master's Guide).

That said, these are outliers and most characters are probably not going to ever actually hit 30 in anything.

The fastest (and not terribly fast) way to get to 30 is via barbarian or monk to 20 (can get attributes to 25 via the level 19 epic boon and level 20 feature) and then use other methods (tomes, epic boons from beyond level 20 advancement, etc) to get it the rest of the way.

The hard rule for an Ability score is 30, just like the hard rule for level is 20, although you can still gain xp and thus gain feats/boons.