My favorite groundcover is cyclamen. Most people grow cyclamen in pots, and even in my area they are almost never seen in the ground. Yet they are very happy here, and spread very fast.
They begin blooming in their second year. In my courtyard cyclemen have become so tightly clustered that nothing else can grow with them, which makes these plants invasive in the true sense of the word.
I bought this veronica (v. montana?) 2 years ago because it reminded me of forget-me-nots, which I love but can't grow here.
The little 4" plant has since spread by tip-rooting to cover about 3 square feet. It looks lovely in early spring.
However, the little stems started going right over lawn edging and into the lawn creating a bit of a maintenance issue: I now have to cut them back a few times a year. It is worth it for me.
I love the exuberance of candytuft (iberis sempervirens), which blooms so profusely in spring that one can barely see the leaves.
Bacopa (sutera cordata) is a lovely groundcover with a long season of bloom, but one plant is capable of spreading to at least 4'. I cut mine back a few times a year.
All my groundcovers present a tradeoff between serving as beautiful living mulch (keeping the soil cooler and supressing weeds) and requiring periodic maintenance to keep them in their allotted space. From now on, as I look for more of them, I will keep this in mind.