Yes, despite some interest once upon a time in having one (now sold) based in Italy. The key issues however are N-registration, a friendly A&P and a place to work. If I owned (or wanted to own) an aircraft in Europe I would focus like a laser on those items and absolutely no way would I register an aircraft in Europe.
That is massively true, but the gotchas for many are exactly these points: you need a nearby friendly and good freelance FAA A&P/IA, and a hangar where you can work. The vast majority of European based owners don’t have either of these. The lack of a hangar alone (most hangars prohibit maintenance) pretty well ensures that an N-reg is going to be a hassle, because if your local Part M has no A&P around, they can’t work on it. So, in a perverse way, it is landowners who run the show (as always 
).
I’ve been doing this since I went N-reg in 2005.
Again, I would focus like a laser on attaining a stable place to work, an A&P friend (people matter) in conjunction with buying an (N-registered and nothing else) plane. If that meant flying out of a farmer’s field or in a neighboring country, so be it. I would choose the type plane of plane to suit whatever solution to those issues was available, not vice versa. I would simply not be willing to submit to coercive and bureaucratic nonsense.
(Eastern Germany looks interesting to me, based on the large ex-Soviet air bases, some of which have GA activity & hangars, low population density and low land values. It would be interesting to hear a counterpoint to that proposition)
Unfortunately, most here in Europe don’t have a choice… but yes the key thing to take away from this is that the “ground stuff” (who can work on the plane, where, etc) is more important than the plane itself.
Lots of people who had a great plane have sold up and often got out of aviation totally because they had ground hassles.
Hence, UL or experimental homebuilt or an old classic Annex I.
Then, if you absolutely need to cruise ICAO-style, 2-3 times a year – rent
Circa 2006 I enjoyed speaking my (broken) Italian for an hour or so with a friendly gent working on his plane in a remote hangar. There are a lot of them. His comment about his base was that the airfield (in south central Italy, an aviosuperficie) was a good one because the police didn’t bother anybody. It would have been a great place to be based, the hangar looked rough but some of the the planes inside were nice. There was a Marchetti S.205 that would’ve been ‘interesting’ to operate from that short field and several nice looking newer Tecnams. My plan at the time was to find such a place to base my normal category N-registered Luscombe, fly it around the country when time allowed (acting mostly as an ultralight, no flight plans or similar crappola and limited ATC contact) and make suitable arrangements for a couple of annual inspections before shipping it back to the US. Obviously I would have been doing most of the maintenance myself and looking for A&P logbook entries. Among other factors, VAT concerns dissuaded me.
@Silvaire, why are you so strongly against EASA registration? In recent years, the advantage of N-reg over EASA has all but disappeared for aircraft based in Europe, especially for sub-1200 kg aircraft (although one has to pick the right national CAA to deal with).
This Q has come up many times. Certainly it depends on one’s particular situation, and airport politics plays a big part. Some threads can be found with a search on e.g.
freelance AND faa AND politic*
It probably is true that for many people the N-reg route has no advantage, but the reasons for this are more peripheral than direct.
No maintenance plan
No ARC renewals ever
Many PMA and STC parts
No contact with CAA
A&P IAs are friendly individuals, typically fair, take untaxed cash payment and do not require any facility or organizational affiliation for anything.
No chance I’d ever in this lifetime consider anything other than FAA registration. My 21 year European based motorcycle is also US registered, for similar reasons: no periodic government inspections of any kind, no limitation to ‘approved’ tires, no approval required for any modifications etc. I’m simply unwilling to deal with that kind of crappola.
