Within the EU you should be able to open a bank account without large problems. Either they offer that already or you can request a basic account if you don't have an account yet in their country (which is not always great, it pays to shop around). Sometimes it takes a bit of work and pressure. I opened a Dutch account right from my smartphone within minutes. Other people I know are customers at an Irish bank (even though they officially state that you need Irish residence) because it enables them to use Apple Pay in Germany.
Most Chinese banks will open an account even on an 24-hour transit stamp (it usually helps to speak some Mandarin). Hong Kong is another possibility and will probably get you better service. Same with Vietnam, Pakistan, and lot of others countries. Some US banks will open an account for non-residents as well.