Lots of countries' healthcare will treat you for free as long as you are there. The problem you need to solve is to not get kicked out of the country, so you need to either have a job, be rich, or have a provable reliable income (like a pension) to get a long term visa, or you need to go back to your plan of stealing someone's identity.
I think quite a few countries would see "kicking you out whilst battling cancer" as a human rights violation, and would only kick you out after your treatments.
I was searching for some precedent there - which brought up Bush eliminating cancer treatments for illegal immigrants in 2007. It's impressive that this war criminal can still disgust me even further.
The thing is that, at least for Americans, their country of origin can treat the cancer as well as any European country, and "I'm too poor to live in my own country" is usually not an accepted reason for immigration or asylum.
There are some countries that have medical treatment visas but those usually require people to pay for the treatment, and while it's order of magnitude cheaper than US hospitals but for cancer is still not exactly cheap.
Touristic visas only last 90 days.
And usually the hospitals will only provide emergency care to tourists. They might be willing to stretch that definition a lot but not to the point of doing things that need to be regularly scheduled.