Not a problem in my world. Not sure I'd even really call that a girdling root. Regardless, someday way in the very distant future, the tree will get old and die. I doubt very much that root will have played any part. My advice would be not to try hacking/chipping/cutting or somehow trying to remove it. It's a big, established tree. Let it be. As they say in med school: first do no harm........

Check the inside of your foundation. Tree roots can enter the smallest crack when they are small and thread-like. Threafter they gradually expand in diameter and can do serious damage. If your foundation is concrete with brick above grade, chances are good that no roots have been able to penetrate. Block or completely brick foundations are far more vulnerable to intrusion. If you want to do something about the root in the photo, you do not need to remove the whole thing - a back breaking project with the potential to do damage as you wrest it out of the ground. Instead, cut the root in two places to leave a gap between the two cut ends. Both sides of your foot in the photo would be fine. Place copper impregnated fabric like that which is used in root control bags around the cut end on the tree side. This will keep the root from resprouting. There are also root control landscape fabrics and I don't see any reason why a piece of copper flashing or copper window screening wouldn't work as well.

The underlying problem, if not excess shade, is root competition. The tree was there first and its needs come first. Its roots have most likely explored every cubic inch of soil. To establish any new plants in that root zone, you will need to manage them as if they're in intensive care. Perhaps set up a drip system for each newbie.

As Babka mentioned, you can use pots and bury them into the soil. But you have to remember to actually turn the pot a couple of times during the season to break the roots that are entering via the drainage holes. I currently have a few pots that are buried. Along with turning the pot, you would need to water more frequently. You may be interested in reading my post. It's about the Fragrant Blue hosta pot that I had buried and never turned for a couple of years. http://forums.gardenweb.com/discussions/3918002/im-a-bad-hosta-momma-fragrant-blue?n=20