After a lot of googling on this - and thinking the crusades might provide an answer - the best I've come up with is Louis IX of France who died in Tunis on the 8th crusade. That's 920 miles 'as the crow flies'.
By furthest from home, I mean from the monarch's capital city.
EDIT
My reason for picking capital city over place of birth is that I am interested in how far monarchs got from their centre of power. And some English kings were born in France (and I guess some other European monarchs were not born in the country they ruled).