2 sessions ago, my wizard found a headband of intellect, and used it to up his spells.
Your Intelligence score is 19 while you wear this headband. It has no effect on you if your Intelligence is already 19 or higher without it.
Now I am close to getting an ability score increase and I want to increase my intelligence. So my question is, can I increase his intelligence from 19 to 20, or do I have to go from my non-modified intelligence?
It is reasonable for you to ask this question, since the rules on Ability Score Improvements simply say:
you can increase one ability score of your choice by 2, or you can increase two ability scores of your choice by 1. As normal, you can’t increase an ability score above 20 using this feature. -(PHB, p. 115, and other locations)
There is no distinction made in here between Abilities that are natural or reliant on magic, so you might object to the other answers saying that you can only increase from unmodified stats. So let's assume for a moment that you can increase from the magically increased stat. What would happen?
Consider the following steps in chronological order:
Your Intelligence score is 19 while you wear this headband. It has no effect on you if your Intelligence is already 19 or higher without it.
At step 4 & 5, your Intelligence is not "19 or higher without" the Headband of Intellect: it would be 17 without the headband (previously 16, increased by 1 by your Ability Score Improvement just now). So the magic of the headband would take effect, and your Intelligence "is 19."
Thus, although the rules (may) permit you to increase an Ability that is already boosted by magic, this particular magic would shift the score (instantly) back.
This is fairly well summed up in the second sentence of the quote, "It has no effect on you if your Intelligence is already 19 or higher without it."
Even if that second line was absent, the end result would not be an Intelligence score of 20; the headband does not increase your Intelligence score, or change some sort of "base Intelligence score", it sets your Intelligence score to 19.
The second sentence isn't there to tell you that it cannot increase your ability score above 19 -- that is given by the first sentence alone. The second sentence is there to tell you that the Headband of Intellect can't reduce your Intelligence score.
You increase your Intelligence score from the unmodified stat.
Confirmed by Jeremy Crawford: Ability score increase only affect values on the character sheet. Headband of Intellect has no effect on the value on your character sheet.
The ability increase ignores the headband. It is increasing the score on your character sheet.
If at any time your Intelligence is less than 19, the headband sets your Intelligence to 19 minimum. It cannot be reduced.
https://www.sageadvice.eu/2015/10/07/multiclass-temporary-stat-prerequisite/
The intent is that your base score, not a temporary score, has to meet a multiclassing prerequisite.
While that tweet is Rules As Written when specifically referencing using temporary ability scores for multiclassing, I believe that it can be applied to this question as well.
From that tweet, we can tell that the designers have intended a distinction between your characters base ability score, and your characters temporarily modified ability score.
While I don't see anything in RAW that mentions Ability Score Improvements applying to the base score instead of the modified score, it would seem from that tweet that the designers did not intend the modified score to be used for level up related things, such as multi-classing and then by extension: ability score improvements.
Additionally, if it were to work in the way suggested, where the ASI increases your Int to 20, then that would imply some very strange rules interactions elsewhere. For instance, if you were to take the headband off and put it on again, your Int would be 19, not 20. That would be a somewhat strange interaction.
Even more strange would be, if you were under the effects of a Potion of Giant Strength (Sets your strength to 21) when you leveled up, RAW would not let you use your ASI to increase your strength, as you cannot increase an ability past 20, and the Giant Strength potion sets your strength to 21.
Both of those would be very strange rules interactions, especially the second one. I'm fairly certain the designers would not expect Potion of Giant Strength to prevent you from leveling. Therefore, given the way that rule would play out, as well as given the Crawford tweet about multi-classing, I believe we can conclude that the ASI was intended to modify your base ability score, not your temporary ability score.
