A lot of people in the US restore their own N-register aircraft as a hobby, and it appears to me that this combined with experimental category aircraft will be the source for our US-based flying chariots over the next couple of decades.
Obviously and as others have pointed out the simpler the plane the easier the project. Don't start with a $20K Cessna 310
One guy I know has over the last 20 years totally restored about three non-airworthy planes, and also built two homebuilts. The last Luscombe started with a 30 year desert stored plane that cost $5K initially, a negligible part of the final total which was probably $35K. He's a single airline pilot in case you're wondering how anybody has that much time available. Like most others doing this work, he'd networked into a community of people doing the same and the work is performed under supervision, logged and signed off by FAA A&P mechanics as appropriate. I bought one of his projects after completion, and its correct that he didn't get any return on his labor - I paid about the sum of his expenses. Its just a hobby for fun and he's happy enough now to see the plane well taken care of.
A lot of parts for certified aircraft can be repaired, or essentially made from scratch under FAA A.C. 4130, so a lot of people doing this kind of work are doing it on planes for which few airframe parts are available. Used parts can also be (repaired and) installed on N-register without great formality, as long as the mechanic judged them airworthy. Among people associated with a given type, there's a lot of parts trading.
Its seems to me that there's nothing in principle that would prevent somebody doing the same anywhere in the world, assuming the aircraft is normally certified on N-register or alternately in another country's sub-ICAO airworthiness regime. Unless I were very well connected I wouldn't beat my head against the wall trying it with a normally certified aircraft on one of the EASA registers.