Direktbanken (i.e. purely online Banks) are an exception but they come with their own set of asterisks and are not a blanket solution as you can't open an account from all European countries.
Deutsche Bank is well known for offering accounts to non-residents. Pretty much any Sparkasse will open an account for you if you walk in even without a German address.
The list of banks that don't offer accounts to non-residents is probably shorter than the other. Granted, these are mostly the ones with the best offers.