Need suggestions: plants that are REALLY hard to get rid of!?

Most things that meet that description are invasive weed. Many of them are beautiful like Bishops weed (Aegopodium podagraria). It grows beautifully under trees, along driveways or walks. It spreads every year and the leaves are great the whole growing season. It's considered nearly impossible to get rid of so it sounds perfect for you. You can actually buy it at a garden center.

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Yuccas and other succulents are good as they will spread themselves quite happily once established. The good thing with a lot of them too is that you just need to break "pups" off an adult plant somewhere and plant into your garden and a good percentage will take if well maintained. From there they require little looking after. In fact anything suited to a desert-like environment would suit

bamboo ,asian bamboo ,any bulb type of plant a(not so much flowers) Lilly of the nile is one even if you think you got the bulbs out from the ground ,guess again ,rhodendron or elephant ear its sometimes called is a royal pain in the butt to completely remove .Mulberry bush (not trees and if you really want a mess get a flowering (produces fruit) mulberry Theres thousands of nasty invasive plants .If you want a nasty tree i suggest, acacia ,any locust tree species ,California pepper The wood is no good for nothing not even fertilizer ,cant burn it its useless and the most invasive root wrecking and spreading tree that at this moment comes to mind is the Brazilian pepper it can be maintained in a bush form but suckers will pop up everywhere a root is or all over the place if left unattended they can develop into full blown trees have fun Do not plant these if your going to be responsible for them no joke

Might help if you said what your use is for this plant and what appearance you want it to have. Ground cover, wildflower meadow effect, or something else? How much sun will the area get? Do you need it to do what it does in the first season, or can you wait a few years? I have a couple of candidates in mind, but without knowing those things I'm not sure if they will work.

Well, yucca comes to mind but I've never seen that propagated from seed. Rose of Sharon hibiscus is tough and re-seeds like crazy and the sumac tree is also like that. Stinging nettle and spearmint are almost unkillable too.

Plant perenials wether flowers or shrubs. Check on the web for different types and designs to plant for your area...

Vinca minor can climb, but it can crawl on the ground too, and it roots like crazy with many branches developing in a hurry. Another one is foti.

Look for species that are labelled INVASIVE.