The first thing that comes to mind is sedum for me. It's dense, low, spreading - and if in a pot, hangs over the edges of the pot. Some of them even bloom. I don't think it needs to be cut down tho. There are MANY different kinds - you'll want one with this kind of growth habit: https://www.naturehills.com/new/product/perennialsdetails.aspx?prodid=1281
as some of them actually grow some height.