I know this thread was started a long time ago, but I joined in order to clarify some things which have been said and are wrong.
1. Maxima will never be able to renounce her Argentine citizenship. It can not be done, even if some anti-immigration politician wants it to happen. She will always be argentinian.
2. Her daughters do not have the argentine citizenship. They can, by birth right, acquire it but to be argentinian you need to have been born in Argentina, regardless if your parents are foreigners (legal or illegal). Therefore, Amalia, Alexia and Ariane are solely Dutch.
3. Reason why Maxima uses her Argentina Passport when visiting Argentina: She can not enter and leave Argentina with the Dutch Passport. I have dual citizenship and can not use my EU Passport when leaving Argentina because I am Argentinian. I enter and leave the country I'm visiting with my EU Passport but when I come back home I need my other passport to enter the country again (the fact I have never lost any of my passports is a miracle!).
In short: she can not renounce her birth citizenship, her daughters, though eligible, do not have the Argentine citizenship, needs arg passport to enter and exit Argentina when visiting her family and friends.
Oh, and even though people in Argentina thinks she is a traitor or whatever for saying she is Dutch and supporting the Dutch football team... they're bonkers. I do not expect anything of her. She is Argentinian, dutch, and future Queen Consort of another country. I'm sure she will always feel love for her country, but she must love The Netherlands the same way.