Just about anything can grow in a container. If you're looking to trade seeds, please go to the Exchange area of this forum or just post "Asian Vegetables Wanted" at the Seed Exchange Forum.
Ron, I've been growing pak choy very well in white plastic bins, about 16X12X8", that I punch drainage holes in. I harvest the greens at the baby stage, at about 3-4".
Ron, the Asian greens seem to all be quite shallow rooted plants. Fliptx suggests pak choy and I think that is a good idea. We grow the dwarf Mei Qing variety and that may be the best choice to allow to reach maturity. If slugs can be avoided, you could really "pack" the pak choy into a pot.
Watering is a real issue here and I've only got so much time so have few containers. They really require daily attention.
Sweet potatoes would make an attractive container plant where appearance is an issue.
If it's not too late to chime in, there were a few melon plants I grew in large pots (cleaned out kitty litter containers) with what I call good success. That would include foo gwa (or bitter melon), mao gwa (fuzzy gourd), delica squash -- is that considered an Asian vege? -- tong gwa as well as tomatoes and hot peppers. I like a good challenge and a lot of diligence on my part with the watering and feeding especially on hot days. I'll never know what I can and can't grow unless I just jump in and do it. And I love watching things grow. Your mileage may vary!