There are others too like carrots, parsnips and beets. There are limits however and there may be a point of diminishing return if "left in the ground all year round". With winter upon us here in the northern hemisphere, this is prime time for storing root vegetables right where they grew, though some things may not tolerate a hard freeze. Winter in my area tends to mean a lot of cold rain, instead of insulating snow, and the soils here can get soggy and result in rotting of root vegetables left in the ground.
"Study books and observe nature; if they do not agree, throw away the books." ~ William A. Albrecht
Horseradish and maybe, just maybe, comfrey if you call it as a crop.
Depending on the climate, garlic can grow wild and even though quality and which part of the plant would differ, it would be avaible year long.
Sweet potatoes and many other plants (like ginger and tumeric) comes to mind for tropics.
Having comfery listed reminded me of Oregon Grape (which isn't a grape at all, it's Mahonia aquifolium). The plant is closely related to goldenseal, and it a powerful antibiotic/antifungal. It's an evergreen plant and so it should be easy to locate the root all year round. There might be better times to harvest it due to roots often storing more nutrients/sugars/chemicals in their roots at certain times of the year, but it should be harvest-able all year round.
I want to say garlic. here in Canada some people plant it in the fall and harvest it in the spring, but some people also grow it through-out the summer I guess? I'm not sure but I think garlic counts as a cold hearty root crop.
All depends on your climate. In wetter and warmer areas it seems that if you don't pull your crop out of many plants that will over winter other places they will sprout anew in what might be called the winter.
You can keep things like carrots in the ground through the winter, using mulch to protect them from freezing (use deep mulch if you have cold winters). However, varmints may find the stored roots and help themselves -- mice and other rodents will tunnel around under the snow. And if the tops of the roots freeze, you may still have some usable carrot under the mushy frozen part, but you'll have to do some digging to get it out of the ground.