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Stemming from this this question:

How exactly does increasing an ability score above 20, by any means, work at all?

Relevant quotes:

Ability Score Increases: Some of these features allow you to increase your ability scores, either increasing two scores by 1 each or increasing one score by 2. You can't increase an ability score above 20. -PHB pg. 15

Manual of Gainful Exercise: ...your Strength score increases by 2, as does your maximum for that score.

Say you have a Strength score of 18, and use a Manual of Gainful Exercise to increase it to 20, can you still not use an ASI to increase it to 22, despite the "maximum" now being 22? In fact, is 20 even the maximum for your ability score when using items; the quote seems to only apply to Class Features.

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  • \$\begingroup\$ I feel like Manuel of gainful exercise permanently raisers your cap, specific vs. General, based on the very valid point you raised. but feel this is a good question to ask sage advice, as I do not Obie of any officiant rulings regarding this issue. \$\endgroup\$ – Garret Gang May 14 '18 at 13:34
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RAW

From strict reading of the ASI, you shouldn't be able to, since it does not say "You can't increase an ability score above your current cap" or something on these lines, it just says "You can't increase [...] above 20".

Rules as (probably) Intended

Now, is it intended? Probably not.

In the PHB, the only way to go over your cap is through Barbarian's Primal Champion feature, which is at 20th level and you are not getting any more ASIs after that.

So one possible reading on this is that it's worded as it is for simplicity, since they were not considering the items on DMG.

The problem is that, from my experience, the DMG is written (more) vaguely (than the PHB), which might even be intentional since it's supposed to be read by the DM and he can decide whatever he wants. This means it's probably implied that increasing the maximum score by 2 (to 22) allows you to increase your stat score to 22 by any mean, including ASI.

A further indication that you should be able to do it is that every mean of increasing your stats says something on the lines of "up to 20" (AFAIK), instead of "up to your cap". This means any feature increasing your cap doesn't do anything, since everything is hard capped in its own text anyway (Ioun Stone as another example besides ASI).

For clarity, what I mean here is that there would be no reason to word it as "your maximum for that score increases by 2" instead of "up to any value" or "not hard capped at any value" or anything like this if it was not supposed to influence any other item or feature.

About your last question

In fact, is 20 even the maximum for your ability score when using items; the quote seems to only apply to Class Features.

The quote is a general rule. This shouldn't matter though, because as far as I know, the items explicitly state their cap (Deck of Many Things' star card caps at 24, Ioun Stones cap at 20, etc).

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Looks to me like however you want to interpret RAW (literal vs what you think is the intended meaning), if you are at 20 and then use the manual, you do indeed raise your stat to 22. So if in doubt, just save the item until your stat is maxed.

From a game design perspective, however, this is just tedious and offers no benefit or meaningful difference. As a DM, I would say that using the manual at any point increases your cap, and thus allows an ASI or feat to bump you up to that new cap.

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  • \$\begingroup\$ +1 because this is how I would handle it as a DM as well. It might not be 100% RAW but feels like what the RAI would be. (Downvoters should explain what is wrong with a question as well.) \$\endgroup\$ – IT Alex Apr 11 at 19:50

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