You can operate a US experimental in Canada, Mexico and the Bahamas without applying for permission as there are standard forms of authority.
Everywhere else requires that you apply for an exemption from having a standard airworthiness certificate.
I gather that extended permissions have been granted to US citizens temporarily relocated to Europe, though the standard permission is as described above.
The amount charged for those exemptions can vary widely from country to country too.
Don't forget that if you get granted extended stay in one European country it doesn't give you free circulation to any others.
You will also have to comply with the insurance requirements of the local authority.