I've studied the rules for Greece specifically. Currently for non-EU visitors, ~$24,000 in a bank account will get you a 'self-sufficiency' residence permit for 1 year. There are other permits available for buying property, marriage, and such, but thats probably not what a nomad would do.
That said, there it is not difficult to overstay the standard 90-day EU visa and simply pay the fine when leaving / on the next entry.
As far as taxes go, I would wager that any American in Greece is only paying taxes in the States. The Greek tax system is notoriously insane.