Excellent answers all. I am familiar with Australia, having lived there for three years, and my daughter just came back from South Africa where she, too, had a life-threatening allergic reaction to something she ate.

I was thinking more of a diagnosis of something longer lasting- e.g., a cancer or something similar. Suppose one just wanted to stay in place and get whatever treatment is available to locals rather than return to the massively expensive US. I have lived in Peru and know some people there. Other than visa issues (which can be handled in a variety of ways), is there something I am not thinking of that would prevent me from just deciding to "ride it out" in such a country?