1. Re-registering your machine on to a foreign register.
2. Organising a maintenance organisation to look after your aircraft on the new register.
3. Finding a foreign-qualified IR examiner to do your IR renewal every year.
4. Finding an AME who can issue FAA medicals as well as CAA ones (Mine does).
5. Face the possibility of explaining to a ramp-checker in Albania whose English is somewhat limited that your aircraft is not actually yours but is owned by a trust in Delaware.
6. Constantly trying to stay one step ahead of the inevitable outcome of EASA banning foreign registered aircraft.
1. no problem - FAA IA lives in the hangar opposite
2. no problem - see above
3. not applicable - rolling currency, plenty of CFIIs around
4. no problem - He does my JAR one and charges me £20 for the FAA class 2
5. not an issue if the paperwork is in order
6. that is the only problem, for no good reason.