I'll add a bit more context. I work in Spain (I'm from Ireland) and only come in/out of Argentina on the 90 day tourist visa at the moment. However, I'm guessing that I would have to spend an extended, unbroken period here before I could even begin to apply for a DNI? But how do I do that if I can only enter/exit every 90 days? Or can I actually enter on some other visa because of my Argentinian wife/baby?
In theory, I could transfer from the Spanish office to the Argentinian office of the company that I work for. However, I have pretty much ruled that out as I have no desire to commit financial suicide and get paid in pesos instead of euro.
The baby was born a few days ago. All good. I will be applying for his Irish passport asap as it carries much more weight than an Argentinian one (will obviously get one of those too).
You can apply for permanent residency now. No need to wait for anything. As soon as you apply online via Radex you will be issued a Precaria which is a temp residency while you case is reviewed. With his precaria you can live here and travel in and out, no need to do the 90 days tourists runs.