GardenersCardiff answers:

It does depend on where you live, but green peppers and peppadews (so the pepper and chili family) can grow as perennials. We actually dig our best pepper and peppadew plants out of the garden each autumn and put them in pots, prune them slightly, and keep them in the basement (there is a window for light) till the next spring, and then we plant them out again. This way we get crops of peppers and peppadews much earlier, and the plants grow much bigger than those grown as annuals.

And tomatoes can also grow on indefinitely. There are tomatoes that are determinate growth that grow till a predestined size and stop growing to produce fruit and die. But there are also tomatoes that are indeterminate growth, the grow on and on, one just has to provide a pest free environment (we did this in our hothouses in South Africa. Especially over the winter months when t was not viable to resow a new crop till spring. We just let our tomato plants grow 5 to 6 metres up on wires and then across the wire, and then down on the other side. When it reached the bottom, we used to let the plants grow up again.)

There is also a bean, the one with red flowers, called Scarlet Runner, that actually forms a sort of bulb in the ground, and under good conditions, comes up year after yea. Again in South Africa, we planted beans that were perennial. We just had the problem that the heat caused the flowers to drop before beans were formed, so the picking season was short and the beans were not as long as they grow here in the North.

In addition to all this info, I found a site about perennial veg:

http://www.agroforestry.net/pubs/perennial_vegetables.html