I believe the terminology you want is that "the augmentation ideal is locally nilpotent." See, e.g. http://planetmath.org/encyclopedia/NilAndNilpotentIdeals.html, although there are surely places in actual literature where this is used. I think an ideal being nilpotent means that the powers of the ideal are eventually zero (and is stronger than just requiring it to be a nil-ideal, meaning it's comprised of individually nilpotent elements). So a nilpotent ideal is one where there is a uniform bound, i.e. I^N=0 for some sufficiently large N. Locally nilpotent means that for any finitely generated subalgebra there exists such an N.

Is that the notation you want? I don't know a name for an augmented algebra whose ideal is locally nilpotent other than just that.

I believe the terminology you want is that "the augmentation ideal is locally nilpotent." See, e.g. Link, although there are surely places in actual literature where this is used. I think an ideal being nilpotent means that the powers of the ideal are eventually zero (and is stronger than just requiring it to be a nil-ideal, meaning it's comprised of individually nilpotent elements). So a nilpotent ideal is one where there is a uniform bound, i.e. I^N=0 for some sufficiently large N. Locally nilpotent means that for any finitely generated subalgebra there exists such an N.

Is that the notation you want? I don't know a name for an augmented algebra whose ideal is locally nilpotent other than just that.