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I've searched through the forums and I haven't seen this exact case before, so here it goes.

The Daemonettes of Slaanesh have an ability that lets them add additional attacks on rolls of 6. I will paste the rule directly from the Warscroll here:

Graceful Killers: Each time you make a hit roll of 6 or more for a Daemonette’s Piercing Claws, that model can immediately make one extra attack. Daemonettes can make one extra attack with their Piercing Claws each time you make a hit roll of 5 or more if their unit contains 20 or more models

Now, bearing in mind that the rules state that you can only ever "reroll dice once", If I roll a 6 to attack, and my additional attack roll is also 6, do I get to roll an additional, "additional" attack?

The reason I justify this is because I'm not rerolling anything. I'm in fact adding an attack. So in theory if I kept rolling 6's I could add attacks infinitely.

What are Dakka's thoughts on this?

Exactly right.
You aren't rerolling because you aren't changing a result.
You get one extra attack dependent on a hit roll.
That attack also requires a hit roll, and there is nothing in the wording that indicates you stop generating further attacks from subsequent hit rolls.
So yes, with a unit of 20+ Daemonettes it might take you a while to determine how many to-wound rolls you'll need...

Yup, none of them are re-rolls. Go nuts and let the attacks continue.

This is great news for me! I'm a little bummed though because my last game I rolled a ton of extra 6's...

If you want precedence too, look to the old 6th Ed Bone Giant for TK.