here is the root.
Can we post more than one picture at a time?

one more try for the root picture.

Here is the stem side shot.

Also, every weed guide I have found on the web site has nothing to offer.
The thought of planting a variety of mint plants to attempt to choke it out has come to mind, but I fear it will just commingle and hide.

Has it flowered? If so, do you have pictures to post or can you describe them?

And where do you live?

I think it's mugwort, one of the artemesia family. It is a royal PITA - my neighbor and I have spent years on getting rid of it.
What worked for us (pretty much) was digging down to get most of the roots and then being vigilant about pulling the little shoots up as soon as we saw them. Like bindweed and some other horrors, the roots that are down deep keep sending up shoots; if you keep pulling the little ones, the rootstock will eventually be exhausted and stop. Every so often, one will show up to my great dismay, so I guess the underground root is still there, but it's like once or twice a year, so no biggie.
Don't even think about composting any of it - I swear, it lives forever in compost.

That does look like Mugwort, "Artemisia vulgaris" an herb reported to have many different curative properties. When crushed these plants emit an odor somewhat similar to whar Chrysanthemums do.
Because of the extensive root system they develop, rhizomes actually, destruction can be difficult. The best way is to keep the plants cut so the roots eventually starve.

It looks like we found it!
Here is the bad news!!

This species is an almost ineradicable weed. Control measures include stubbling, early autumn plowing, crop rotation with bare fallow, mowing on fallow grounds before flowering and use of herbicides if necessary.

I guess I'll just have to tear the whole garden bed out and sift out the weed roots.

http://plants.usda.gov/java/profile?symbol=arvu

THAT WEB SITE SAYS THAT IS IS NOT IN ILLINOIS, MUCH LESS IN THE JOLIET AREA!!!

I WILL BE PRINTING THAT PAGE OUT, AND IN THE MORNING I'M GOING TO PLACE IT IN THE FLOWER BED!!

I'M SURE THEY WILL RELOCATE ASAP!!!

"I'M SURE THEY WILL RELOCATE ASAP!!!"

If only that would work!
Good luck with the mugwort - it's a job and a half but you can do it.

Well, it's been a year and I took a week end and just tore up the first 6 inches of soil and sifted it by hand.
All spring I have been digging and pulling.
If I see a shoot come up I do not stop digging until I find the root that supplied it.
While I was at it I pulled the crab grass as well, but it is more prevalent than the Mugwort, "Artemisia vulgaris".
I tried to sell the neighbor on taking it as it was used to brew beer before there were hops. He is not interested.

A few years ago I did get it all out of one small bed even though it was firmly tangled in a root ball of a bush.
I do believe that is where it all started from, the bushes planted by the builders landscaper.
How better to get rid of a weed tangled bush?