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Hexblade's Curse has several bonuses applied to when an enemy suffers a damage roll from you.

Does damage from something like Armor of Agathys, or an unarmed strike, count as a "damage roll", despite not having any dice?

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It's not a damage roll unless you actually roll

There is a word the rules use when they talk about damage that is not (necessarily) rolled: "damage". However, the game defines a damage roll to be something that includes a...roll:

Damage Rolls

Each weapon, spell, and harmful monster ability specifies the damage it deals. You roll the damage die or dice, add any modifiers, and apply the damage to your target.

Note that the rules for damage rolls specify that they roll "the damage die or dice" not "any damage die or dice". They are a non optional part. Compare this to " add any modifiers" which shows that modifiers are an optional part of a damage roll.

While every ability specifies the damage they do, damage rolls explicitly require a roll to be made as part of them (even though static modifiers also sometimes get added).

This also makes sense with 5e's policy of using natural English. If you say something has a roll why wouldn't it include a roll?

Hexblade's Curse says:

You gain a bonus to damage rolls against the cursed target.

Like you said, armor of Agathys does not involve a damage roll and thus the bonus would not be applied to it.

As always, the DM at your table can decide they do not like this and rule that the bonus does apply at their table, but there is no reasonable way to ignore the significance of the word "roll" from a rules perspective.

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No.

D&D 5e uses regular English language, not "lawyer English" (like previous editions). Imagine you walked up to a random person on the street who has never played a pen & paper game and asked them

If you have to choose a number, but you take a fixed value instead of rolling dice, have you made a roll?

Would they answer "Yes"?
Skipping those people who'd go "dafuq?" and those who would troll you, you will most likely get "No" for an answer.


Conclusion: static damage that does not make you roll is not a damage roll.

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