It looks like broadleaf dock, also called bitter dock. It does have a very deep taproot and will regenerate from pieces left behind near the surface. If you don't want to use Roundup on it, probably the next best thing is to use an asparagus knife or shovel to keep cutting the crowns off several inches below the surface until the roots eventually die. That may take awhile, because it is a very hardy and long-lived perennial. The flower stalk is tall, so probably you won't see any flowers if you are mowing over it -- and a good thing, too, because the seeds can sprout after lying dormant for 50 years. So if your lot was an open field a few decades ago, or you have had topsoil that contains dormant seeds brought in, you may see some new seedlings appear from time to time.

I don't mind using Roundup and will buy some this weekend! Thanks!

FYI - roundup will kill the grass too. I would get the concentrated stuff and an old small paintbrush, cut the dock at the base and paint the roundup on the cut ends. This will kill only the weed, and none of the surrounding plants.

Also, if it really is dock, you can eat it when it's young - it taste sort of lemony and isn't bad in a salad. :)

24d (WeedBGon) will kill this stuff w/out harming the lawn.