Well I have no bank account here but I was thinking about making one and the information I gathered is the following
1.) Apparently applying for the rentista visa requires a bank account here. So there is definately no law forbidding to open an account without DNI. Banks can obviously demand it anyway since it is their decision which extra requirements they demand.
2.) By law, a CUIT should be sufficient. Don't mix it up with the CUIL (the latter always being tied to a DNI). You have to make police verify your adress and that you are living there (can't remember how it was called, something with "domicilio"). With the stamped/signed sheet of paper you get from them, you can go to another authority (ANSES?) to get a CUIT (or a CDI and then a CUIT)
The CUIT is a kind of tax identification number. As far as I understood, CUIT is geared towards businesses while CUIL is for employees.
3.) As you see, I am not remembering everything in detail but the terms should be good keywords for a search on the details of the steps. People reported that it was a matter of luck whether a bank allowed opening an account with just CUIT + passport or not. I.e. same bank, different branches, 4 say no, one says yes.
I researched the topic on this forum about half a year ago, I hope that there weren't any big changes in legislation (or in the corporate guidlines of the banks) since then.
PS: Another european bank seems to be BBVA Banco Francés (apparently part of a spanish group and not related to french banks). Unless I missed it, only HSBC, Santander and Itau had been mentioned in this thread.