Hey JoeCoe. Curiosity is great....Im just bringing fact to this topic so everyones curiosity with regard to this is satisfied. I know alot about this because Ive been all over this and back again.
Any extra license is valuable to have for sure. No denying that. But generally, if an AC is registered in a certian country, you have to have a license from that Country to fly it. For eg, Vietnam Airlines have AC registered in the Seychelles. But to work there you first have to sit and pass the Seychelles Air Law exams. In Europe, its no dif. You want to fly Euro airplanes, you gotta hold the Euro license, and that means sitting all the exams. If you have the right to live in Europe like I do, then its possible. If you don't have the right to live there, its not.
Secondly, about your N reg AC...you cant fly N Registered AC in other countries unless by some techincal agreement. Same goes for Euro. For eg: An N reg AC cant fly to europe and compete for business with the Europeans. Thats called "Cabotage". You couldn't fly down to Seattle, pick up some passengers, and drop them off in San Fran. That again is Cabotage. There are exceptions, like in private company jets picking up company personnel, and things like that.