BTW -- I'm not advocating anyone do what I do. I don't worry about my Visa situation because I'm marrying my Argentine boyfriend at the end of the year and then will have to start the DNI process.
In my experience, the more integrated into Argentina you seem, the less hassle you're going to get from the authorities. Also i'm Canadian, not American, so I think I'm automatically get a leg up on the Yanks, though with Obama you guys might be liked again!
I speak pretty fluent Spanish, though with a shocker of an accent, and the officials seem to appreciate that. I don't try and pretend to be naive about my situation. I've usually had very friendly officials both at migraciones and customs -- customs officers usually flip through my passport and ask if I have a boyfriend, and I'll reply something like "Si, me enganchó!" which usually gets a bit of a laugh.
However, when I go to the airport, I'm usually surrounded by my Argentine family -- I can never leave town without at least my boyfriend being there, if he's coming too then it's usually + mum, + dad, + one or two siblings, + 1 year old niece etc. It's quite the scene. I don't look like I'm some tourist, I look like I'm pretty integrated, and I think that helps a lot.
Or maybe it's just that I have boobs and I'm blonde... hmmmmmmmm