Anyone from anywhere can enter Ecuador with a T3 (tourist for three-month) "stamp." This isn't a visa. Your passport is merely stamped when you come into Ecuador and it's good for three months.
In order to stay for longer than three months in any one-year period, you'll need a 12-IX tourist visa, good for up to six months in any one-year period. The following four posts describe the process, step by step, as I experienced it (though Part Three is Members-only for reasons I reveal there). It takes some effort. But depending on how far away from an Ecuadorian Consulado General you live, it's relatively painless.
So now I face the challenge of establishing residency. Read Visa Part Four for a brief description of why that might be a bit dicey for me.
However, I'll be collecting as many residency stories from gringos as I can while I'm down there, so they'll all appear here.