Perhaps I should say
what I need this feature for. I am planning for a low (9-12") stone wall
in my backyard. I am thinking of curving one end of the stone wall around and
tapering it off out of sight. I need this feature (overflowing flowers from a
toppled pot) to further disguise that end of the stone wall. A further plant
right behind the stone wall and trailing down the stone wall creates a 2-tier waterfall of flowers. Alyssum
(Lobularia maritima) would work well because of the small dense flowers that
will naturally grow out of the mouth of the pot seeking sunlight (I love the fragrance too
:-)) but it is unfortunately an annual. I am looking for a hardy (USDA zone 4)
perennial for more constancy and opportunity for making improvements to that end of the stone wall from one
year to the next.

Although, I would rather not
restrict this thread to what I just wrote in the previous paragraph.
Any plant suggestion that could create 'overflowing flowers from a toppled pot' is welcome :-)

(And yes, I did think about spreading and trailing evergreen ground covers for essentially the same function, but that would be for the conifer forum ... one thing at a time ... I still have 4-5 months time ...)

How about lamium? A quick spreading work horse...

The
particular species of Lamium shown in the photo is too coarse for a smooth
flow, perhaps there are other Lamium species with finer textures. The Lamium genus has
both perennial and annual species, BTW.

Anyway, here is a comparison with
Alyssum which is however an annual, if Alyssum were a perennial, it would be
perfect for my application.

Erodium reichardii. Some of the alpine campanulas, saxifrages. Will have a longer think tomorrow (it's my bedtime now).

No perennial will have as extended a bloom time as the alyssum will plus it will reseed to come back year after year, so very nearly a 'perennial'. If you do want a similar look, consider candytuft, Iberis sempervirens. Or with notes to the above, Campanula portenschlagiana or C. poscharskyana :-) Again, these will not have that very extended - all summer - bloom time of the alyssum. But a pretty long one for a perennial.

OPs concurrent post in the conifer forum

http://forums.gardenweb.com/discussions/4343493/dwarf-conifer-groundcovers?n=10

Yeah, I posted in the conifer forum as I already mentioned in a previous post in the above, to discuss the equivalent effect using conifers. The conifer forum has the pertinent expertise, but so far at least neither conifer nor perennial are as attractive as the annual Alyssum from an overall perspective. Have not made a decision yet.

Where I live, planting alyssum is an invitation to the small black plant lice, who can do much damage in just a day. Wish it wasn't so as I love alyssum.

How about campanula Blue Clips? With deadheading mine bloom spring through fall.

Yes, sure it is an option, it has a rougher texture than the Alyssum, but the color blue may be nicer than white as in the pic below, depending on color coordination.

As I alluded to in an earlier posting, I wish that there was a way for a mix planting like in the original photo, but better. The red in the original photo is way too blotchy, little red stars spread in the white stream would be much more beautiful. However, if the red plant wins in one location, it tends to make a blotch at that location. If anyone knows how to spread little red stars (or blue or other color) in the white stream, that would be great!

Campanula carpatica 'Blue Clips'

The only perennial I can think of with continuous bloom is corydalis lutea. It might be a bit too tall unless a larger pot is used.

Pseudofumaria lutea formerly Corydalis lutea seems to like a woodland setting or edge of woodland as in the photo below.

C lutea usually grows out of walls where I live. But it does enjoy some shade. Campanula portenschlagiana grows in the same type of habitat. Neither seems very successful in the ground in my climate. Both look at their best in a vertical situation, not in a flat bed.

I would spill some 'Angelina' and other sedums out from that clay pot.

Thanks for the many, many suggestions! There are sufficient options for making a couple of 'streams' and changing the 'streams' from one year to the next :-)