score:2
I have completed the trip and can say that leaving a car behind without receiving money for it is easy as you donate the car to the foreign government. It however depends from country to country how easy this process is. My advice would be to consult you Embassy about this matter in the country you wish to leave a car.
Upvote:3
Every country is happy with export. Just make sure that it is considered an export - you'll have to deregister your car in your home EU country so you don't have to pay insurance and road tax anymore.
But it mean you import your car to the country you're going to shrot it. It might be necessary to pay import taxes, which might exceed the actual worth of the car.
The scenario you might consider is to declare that your car got broken abroad and therefore it was trashed. You should inform yourself what procedures are required in that case. Depending on the laws of the country you come into it might be just the best option to crash you car against a tree a few meters from junkyard and get police protocol to proof in your registration country the car was damaged in an accident.