Just to clarify Rich's question and Amigo's comment regarding when I last opened a free Caja de Ahorro I have not opened an account in the last couple of weeks.
The last account I opened was in March, 2016 before the new regulation from Banco Central came out. So at the time I opened the account it was not free. But thanks to the new regulation it has become free, now. This last account was with Banco Macro. But it was the second try with Macro. The first time (different branch) they refused to open an account for me (they wanted the proof of income that Steve mentioned, which I did not have at the time). From what I understand private banks are interested in selling you a package with checking account and credit card and so on. Somebody without proof of income will not qualify for such a package. They are simple not interested in opening an account for you.
I had no issues opening an account at the state owned Banco Nación back then. They did not ask for any proof of income. Only downside with Banco Nación are the long lines. So again based on my (limited) experience I recommend to go to Banco Nación, if you have no documentation for your income. Also if you should run into problems opening the account - try a different branch.