What bureaucracy is scaring you? The process of obtaining an EASA license based on your FAA license if you want to fly EU-registered planes, or the flying in Europe itself?
For many of us, it is both; I have been led to understand that there are a reasonable number of N-Reg aircraft in Euroland that us FAA-cert holders can rent and fly, but I have never done anything about that because none of my trips across the pond have left me with enough time to pursue flying. A friend and I have noodled around about the idea of flying to Europe from north America over the old "Blue Spruce Route" (Canadian maritimes, Greenland, Iceland, Scotland), but time, money and abject fear of frozen drowning death have tamped that down.
Well, I do not have the knowledge what the US or EU require to operate an N-registered plane that is based in the EU. But the flying is very much doable. Assuming that we are talking about a SEP and (mostly) VFR, there are plenty of airfields that are available. Landing cost vary from about €12 to €60, with the bigger international airports very expensive (hundreds of €). If you want to control the cost, always look in advance if handling is mandatory, and is so, what these cost are. These can be, and often are, rip-offs. You can also keep an eye on where to refuel. It is more expensive than in the US, but there is much to see. We have been going on holiday with a SEP for many years, and I would argue that it is a viable travel option (we have visited many places by plane, and there are many more that we have not yet visited, especially for example in Scandinavia, the Baltic states, Ireland).
Flying my own N registered plane that I would bring from the USA.
It sounds like once you figure out which fields to look at and who to call, there are viable options. I am looking forward to learning that, because I'd really like to explore Europe by plane one day (all VFR). Thank you.