I've grown eucalyptus from seed. It was very easy to start, but I'm not sure it would overwinter outside in your climate. Mine can stay in the ground for years and grow very well (up to about 10 feet) then one hard winter will do it in.

You could grow it, most of the fragrant ones are warm growers, globbulus, smithii, citriodora, nicholi....
If you have a green house it would do well. They grow EXTREMELY fast, about 8 - 12 feet per year. Less so in pots, but still fast.

Here in the north, it is, of course, strictly an annual. But, from seed started indoors about 8 weeks ahead of transplant at your last frost date or just a bit before, you can still get impressive growth in good, rich, well-drained soil with a lot of watering and fertilization. Six feet of growth with considerable branching is not out of the question at all, assuming you have about a 5 month growing season outdoors.

I personally enjoy citriodora, cinerea, and globbulus as the fastest growers. Some of the less common types can be fun, too, but frankly, I've never had much luck discerning the scents claimed for some of the kinds -- nicholii is claimed to smell like peppermint, for example, but frankly, it just smells like Eucalyptus.

For best results starting seeds, use grow lights and heat mats.

One good source for a couple of the species is Sandy Mountain Herb Farm in Alabama. They carry citriodora and globulus.

Dennis
SE Michigan

And if it's likely to grow well, remember eucalyptus trees can get VERY large - and don't plant it anywhere near a building, or park cars (or children) under it. Eucalyptus trees are famous for their falling branches once mature. They're the first to succumb in a bit of a breeze.

I grow Lemon Eucalyptus Eucalyptus citriodora here in south Texas, zone 9, and it does great since it is zoned 8 to 11. It has grown from being about 10" in a 4" pot to about 15' in 3 years in the ground. I just love it. You could try growing it as a container plant, maybe.

And here is some information on Eucalyptus cinerea which is also zoned 8 to 11.

More information on The Herb Companion.

I would love to find a small live Eucalyptus plant to grow indoors. The more fragrant the better.

Here is a link that might be useful: My website

Hello started eucalyptus cinerea under lights in late feb.Got some germination,but now it is april and they seem to be growing very very slowly(still under the lights)Any ideas?I am using a water soluble fertilizer.

So if I wanted to grow lemon eucalyptus in the ground as a sort of annual bush, how is this for a plan: at first frost, pot a small cutting for next year, yank the rest, and dry the leaves?
Do I want some root on the cutting? I see comments that seed is tricky, but not much on cuttings.
Can I use the dried leaves for mulch, like under a hammock?

I hate mosquitoes, but I hate DEET too.