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I got a Tiger 800 for almost exactly the same reasons as you...rode it up and down to Greece this year, through Italy, all over south and west Switzerland, France, etc etc, and put loads of miles on it in the UK too. Triple engine not for everyone but I love it. Very worthy for consideration! And a 2nd hand model less than 2 years old is easily within your budget, especially as new models just released.
So...
I quite fancy a bike to do adventures on, but realistically the chances I'll do epic off-roading on it are low. I'd like to ride down to see friends in the German Alps, whizz up to the Highlands, climb a hill, whizz home. That sort of stuff.
I don't want it massively heavy or massively fast and I'd like to spend £5-7k ish.
Would I be daft to get a little adventure bike like a BMW F800GS or a Honda 700 Transalp?
Should I just go for a proper sports tourer?