Are you a citizen of a visa-exempt country? If not then you need to refer to the terms of your visa. For the rest of the world, the 90 day permission one will have received may be renewed one time provided one doesn't stay more than 180 days in any year. Some visitors however take a very flexible approach to visiting and those with long term plans even more so. Use the search box on this site for more information.
If you haven't been to Argentina before, recently and you only want to stay until your AirBnB reservation expires there should be no problem with renewing your permission or not renewing and then paying a small penalty when you go
Again, there is lots more information by following onsite search
As soon as you do not show up at Immigration to apply for a renewal, you can stay for ever. I had a client who was here 25 years before she applied for citizenship.
You do not become illegal.
You pay a small fine when you leave.