Common bindweed (Convolvulus arvense) may be the toughest weed in the world. Now the aerial shoots don't look like too much, a slender vine with it's little pinky-white morning glory flowers, but it forms a tuber deep under ground from which it can resprout forever as best my efforts can determine. So you think, OK I'll dig the sucker out, but it's tuber and roots have been found 30 feet below ground! It will survive any thing short of a direct hit nuclear blast. So you decide to hoe the heck out of it, but new plants can regenerate from bits of rhizome as little as 5 cm long; think about sorcerer's apprentice here. And of course numerous shoots can arise over a considerable area from one rhizome/tuber/root complex way below, so there may only be one bindweed in my garden. Even repeated applications of herbicide will not kill it even though there is some meager satisfaction is seeing the aerial shoots die. New ones will soon sprout, and it can keep it up longer than you can. Some people have actually suggested that bindweeds may grow straight through the Earth and their shoots on opposite sides of the globe are actually what holds the whole thing together. So having just weeded, they'll all be back when the Phactor returns. A close contender may be the trumpet creeper. None has ever grown in our yard in memorable history, but shoots keep arising from some deep under ground root system, and they've been pulled and herbicided every now and again for 10 years now. Any other nominations for really tough to kill weeds?