I'm not too sure if I'd have a pet name for
Anastasia, either. I wouldn't be calling her
Anina all the time, it would just be a little term of endearment. :) But I can see the other poster's point that it might be a bit too pretentious.
As for the other pet forms, there's not a lot of choices (that I like, anyway):
Nastya,
Asya, Asulya, Asyanya, Nastenka, Nastentsiya, Nastukha, Nastulya, Nastyusha, Tosenka, Nastyushka, Asyusha... :S
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We know what we are, but know not what we may be.