Of all the vegatables, only two can live to produce on their own for several growing seasons. All other vegetables must be replanted every year. What are the only two perennial vegetables?

There seems to be more than two. There are 20 according to
http://www.smallfootprintfamily.com/20-perennial-vegetables

1. Berry Bushes
2. Asparagus
3. Rhubarb
4. Artichokes
5. Kale and Collards (usually grown as annuals)
6. Garlic (usually grown as an annual)
7. Radicchio (usually grown as an annual)
8. Horseradish
9. Lovage
10. Watercress
11. Bunching or Egyptian Onions
12. Daylilies
13. Good King Henry
14. Groundnut
15. Jerusalem Artichoke
16. Ostrich Fern
17. Ramps or Wild Leeks
18. Scarlet Runner Beans
19. Sea Kale
20. Sorrel

Rhubarb certainly seems to be classified as a vegetable, though it is used as a fruit.

https://en.wikipedia.org/wiki/Rhubarb

"Some well known perennial vegetables from the temperate regions of the world include asparagus, artichoke and rhubarb. In the tropics cassava and taro are grown as vegetables, and these plants can live many years." While I would probably consider rhubarb as a fruit, due to the way in which I am accustomed to eating it, (and of course, tomatoes are generally referred to as vegetables despite being clearly the fruit of their plant), artichoke and asparagus are probably the intended answers. The tropical ones should not be overlooked, either.

There are also a number of leafy plants that have been regularly consumed in the past, but not so commonly now, such as sorrel, that are perennial.

https://en.wikipedia.org/wiki/Perennial_vegetable

In terms of perennial vegetables, there are several, depending on how adventurous a cook you are.

Horseradish, Rhubarb and Asparagus would be the most familiar.

Less common ones are Cardoon and Artichoke (hardy only in very mild winter regions like California and the Pacific Northwest), Sorrel (greens are used like spinach in various soups mostly), Welsh Onion (green leaves are used in salads in the spring), and then the ones that ride the line between being a herb or a vegetable, like Parsley. There are several other more obscure ones that are now seldom used.

http://www.perennials.com/seeanswer.html?item=145&type=Vegetables&start=0