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The Samurai fighter archetype (from Xanathar's Guide to Everything, p. 31) gains the Rapid Strike class feature at level 15, which says:

If you [...] have advantage on an attack roll against one of the targets, you can forgo the advantage for that roll to make an additional weapon attack against that target ...

So, let's say I am invisible as per the greater invisibility spell, or the enemy I'm attacking is blinded, or something else; either way, constant advantage on all my attacks.

At this level, I can make 3 attacks via Extra Attack, so let's say I hit an enemy twice, with advantage, but then for my 3rd attack, I decide to forgo my advantage for an additional weapon attack, as per the Rapid Strike class feature.

Clearly this 3rd attack will not be made at advantage because I deliberately chose to forgo advantage as per Rapid Strike. But would the additional (4th) attack I make still be made at advantage? Or is it implicit that this additional attack is also forgoing advantage?

RAW, you only forgo advantage on that roll

You've quoted the only relevant rule, which seems pretty clear:

[...] you can forgo the advantage for that roll to make an additional weapon attack against that target [...] (emphasis mine)

You only need to forgo the advantage on that specific roll to gain the benefit (an extra attack). You can roll the additional attack with advantage.

However, arguing rules as intended (RAI), I'd apply it to both rolls

The text of the ruling, (specifically the use of the word 'forgo' and the statement that 'you learn to trade accuracy for swift strikes'), implies an exchange is being made by the player. They are swapping the benefit of rolling one attack with advantage to instead roll two attacks. This is always a beneficial trade for the player to make, as it could allow them to hit twice, rather than once, however, an exchange has been made.

Following RAW, and assuming an ongoing source of advantage (which is pretty common - especially with the Samurai's Fighting Spirit class feature), nothing would be forgone by the player. They would be swapping one attack with advantage for one attack without advantage and one attack with advantage. That's not a swap - it's just a net gain.

For this reason, I would rule that RAI both attacks should be made without advantage.

You'll gain the advantage on the gained attack as usual

The wording says "you can forgo the advantage for that roll" (emphasis mine), making it clear that only a single roll is affected by you forgoing the advantage. Therefore in your example, you would make the third attack roll without advantage, but the extra attack roll you gain would be again with advantage.