I went to a college in the United States where admissions are need-blind (they don't consider how much financial aid you'll need in their decision to admit you) and that offers full-need aid (once admitted, they will meet any financial need you demonstrate). I was an international student, so the aid was not in the form of a loan, but a straight-up grant. I basically ended up paying nothing to go to a college that normally charges $60k+ a year. So if you're not American, this is a possibility. If you are American, I understand that most (all?) of the financial aid is in the form of federal loans, which you may or may not want to incur.

Wikipedia says there are only seven US universities that offer full need-blind aid to international students. There are many more that are need-blind and full-need for US students, although this will probably involve loans. That Wikipedia page also lists four non-US universities that offer need-blind and full-need aid to all applicants. If you are American, applying to one of those may be a better bet, because you might get a grant instead of a loan. I've heard good things about the National University of Singapore.