I'm in the same situation as the OP, with a eucalyptus that I really want rid of. It was just a small tree when we moved here 19 years ago - if only I had known then what I know now, and got rid of it!
Several times over the years, DS has climbed it and sawn off all the branches that are thin enough to be sawn by hand, most recently this summer, when he left it without a single leaf. It only took a month for it to start sprouting new foliage. Then there was the problem of getting rid of everything he had removed - I sawed it all up and burnt it in the garden. It took me all day, as I knew I had to keep the fire small if I didnt want to cook all my nearby vegetables!
I've made up my mind that in spring it is going to be killed, and have been researching how best to do it with strong weedkiller/treekiller. If it works, I will be left with a skeleton which will have to stay put for as many years as it takes for it to rot. I think it would look very attractive with a climbing rose or something similar growing up it.