I've got a friend who just walked into a Wells Fargo branch, and basically all he had to do was to fill out a W8-BEN form. No SSN, no ITIN. His passport number was apparently a perfectly fine substitute.
This was some years ago though, but assuming not much has changed, I don't think it'd be too difficult.
You are also required to disclose any such US accounts to your european financial institutions that you work with.