You could probably braid the trunks of willow.

Aesthetically, braiding the trunk of a tree that is grown for its contorted trunk and branch shape sort of defeats the purpose to my mind.

I don't know what might work, though I would be more likely to try this with a large shrub rather than an actual tree, so this question might be appropriate on the shrub forum. I've seen Wisteria vines braided to create a "trunk", and then the plant pruned to create a tree form but I don't know that I would recommend Wisteria since it tends to be a fair amount of work to control both suckering and seeding around.

Any
specific willow cultivars? I think pliant trunks are needed for beginning the
braid with at least 3 saplings. I also like dwarf cultivars for keeping the
resulting tree at 5' height or less, perhaps that necessitates an upright (not
weeping) growth habit for pruning off the growing tips.

As to corkscrew trunks, I was not thinking of braiding them, but letting them naturally intertwine for increasing the effect. The highly contorted Harry
Lauder's Walking Stick Corylus avellana 'Contorta' is unfortunately not reliably hardy in my area, and the other cultivars seem to contort (much) less. Hence, my thinking of intertwining multiple corkscrew trunks. If that works, one could control the amount of contortion in the overall tree by intertwining a higher or lower number of corkscrew trunks.

As to braiding multi-stem shrubs, yes, that may work also. Maybe a follow up in the shrub forum is appropriate, depending on how this thread goes.

Another aspect to discuss is fusing the trunks (or stems), perhaps inducing it by scraping off the bark where the trunks touch.

You might enjoy searching the term "pleaching". Interesting subject. Interesting hobby.

I don't know willow cultivars that well. What about pussy willow?

My
present focus is on a single plant (small tree or shrub or at least give the visual impression of a single plant) for the longest
possible interest. Pleaching, as the term is usually used, involves multiple trees, but perhaps pleaching techniques can be used to create winter
interest for the bare branches of that single plant. Any specifics?

Ideally you want something that is flexible when young and has a smooth trunk. I think willow tends to be flexible Pussy Willow has those fuzzy buds for added interest and is small. If winter interest is the goal, what about winterberry holly?

Ironwood would be gorgeous, but I don't know enough about the consistency of the wood. You could probably do it with bamboo, but be careful to avoid an invasive species.

Pic
#1 in my original post may actually be Pachira glabra which develops a thicker base at a younger
age although it is allegedly often sold as Pachira aquatica.

Caution: The two species
require different care, please google if you are interested in buying a money tree. I can no longer edit the original post :-(

I
think we can benefit from the shrub forum, I opened this thread over there http://forums2.gardenweb.com/discussions/4328963/m=23/braiding-shrub-stems-into-a-tree-trunk

I will see if I can get some pics of the plaited forsythia. I thought I had some that I posted on another thread, but I cannot find them.