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F800gs has 21inch front wheel so would be a bit compromised on road. But still mor than capable. If you don’t want to do much more than 100mph and don’t plan on touring 2 up fully loaded a bike like that would be fine. A good choice tbh.
Could consider a ktm 990 adventure. Honda Africa twin (newer). Triumph 800 adventure.
I’ve done some 800 mile days on my 990 smt. It’s good.
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I have a Transalp, not adventured on it, but it's happy bombing along at 80-90 and can be wound up into triple figures easy enough (even with my 18stone lump on it)
I get about 150 miles before needing to fill up, but that is after 20 mile each way commutes, better mileage on longer hauls, maybe 170+?
Upright riding position suits my 6'6" frame better, no issues with buffering over the screen, plenty of room for me to crouch/tuck in if I want aeros.
I've not really ridden that far in 1 go, but I never found it uncomfortable.
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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?