Looks like this person didn't get an answer 4 years ago, but I'll repeat for him. I've gotten by for the past two years without a CUIT number, but I have recently contracted some work, and I would be better off with than without. I feel bad asking people to borrow theirs or pay for a sheet. When I tried to get one 2 years ago I was turned down for legitimate reasons (on a tourist visa). Just as a side note, they were quite nice about it at the AFIP office.
However, I know people on a tourist visa who have gotten CUIT numbers and gone on to successfully print invoices. I talked once with someone who said it didn't matter which AFIP office you go to, but I had been under the impression that I had to go to the one that corresponds to my address. I was hearing that sometimes people got lucky (because AFIP employees weren't following the rules, I suppose) but most of the time they didn't.
Other than suggesting I get a work visa (which won't be happening in my case unless there's some possible process here I don't know about), any advice?