Operating N-reg planes is not bis issue, the only thing you need is the respective permission to enter each country (usually issued for 6 month, the old 1 year permits are qute rare nowadays) - but, as there is no country authority "Europe" you have to do this for each separate. So, travelling across Europe with N-reg can be tricky.
US Experimental is even more tricky, as you have to check each and every countries requirements to allow N-reg Experimental to operate in their airspace. Some require noise certificates, some certified inspection, or don't allow that specific experimental to fly their country. Usually permits for Experimental are limited to 3 month, so if you try to do such, be prepared to fight a huge load of paperwork and frequently. You may not need a big purse such as Red Bull, but it is a costly business.