Does anyone know what this is growing next to my hosta, on the right?
I planted some bare root Lily of the Valley in this bed a few years ago, so now I am scared to pull anything big until I identify it!

Looks like creeping jenny.

Next to your hosta there appears to be a very healthy Rumex obtusifolius...common name broad leaf Dock.
Suggest you remove it including ...all the root...asap.

Creeping plant Lysimachia nummularia...common name creeping Jenny.

I know the creeping jenny is indeed taking over. I look at it this way- least I don't have to mulch this bed!

Silversurfer, thank you for the weed ID! I keep hoping that Lily of the Valley will come up, but nope!

By now your Convallaria majalis...common name Lily of the Valley should have grown , flowered and started to die back.
Suspect your bare rooted plants died.
When happy they expand to form large patches.

If you find someone who has it they will be pleased to give you some fresh roots.

Sorry Codie--I just looked at the first picture, and thought you were talking about the creeping Jenny! It is such a pretty bright green groundcover.

its ok, pinkruffles!

Silversurfer, I'm sure you are right, the Lily of the Valley is most likely long gone. I bought it in a plastic bag from a big box store. I guess I learned a lesson there.