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We show that asymptotic (valued differential) fields have unique maximalimmediate extensions. Connecting this to differential-henselianity, we provethat any differential-henselian asymptotic field is differential-algebraicallymaximal, removing the assumption of monotonicity from a theorem ofAschenbrenner, van den Dries, and van der Hoeven (arXiv:1509.02588, Theorem7.0.3). Finally, we use this result to show the existence and uniqueness ofdifferential-henselizations of asymptotic fields.

Differential-henselianity and maximality of asymptotic valued differential fields