My overstaying process went so swimmingly that I am shocked to see people are now having trouble.
I moved here in 2011, had no reason to leave so stayed until May this year without going in and out constantly and getting stamps. In May this year I wanted to head back to the UK, so instead of go through nerves at the airport of will they/won't they, I went to the immigration office and paid the fine there about 9 days before I left (I think the limit is 10 days). That was my 300 pesos paid and a form to be signed at the airport. At EZE on the day I went to the booth, he barely glanced at the paper and waved me through.
Coming back two weeks later I thought that possibly the fact I overstayed a good three years might cause me problems. The guy asked me what I was doing here (in a completely non aggressive or judgemental way), I said I was here to get married, he seemed delighted by this and told me good luck and in I went. In fact the whole process was very serene, my bags were absolutley loaded with "stuff" and I just sailed through.
Anyway, perhaps things have changed since then, which would be worrying for anyone up to the old tricks. Or, perhaps there are some things to consider (yes, here I go with a it worked for me so may work for you bit).
Did me going to the immigration office before the day of the flight help? Certainly, having the paper in hand resulted in the fellow simply looking at it with a glance and moving me on, it was not at all busy with no line behind me at the airport either. I wonder (to myself but out loud here) whether the fact the paper came from someone else took responsibility away from him so he didn't care.
I guessed that even though I had overstayed 3 years that it helped me more than if I had loads of stamps. I am not even sure they check the date as they look so uninterested, perhaps its the number of stamps they look out for.
Or perhaps it is that they are clamping down on this and I sneaked in during the final days of the good times.