I was expecting to see a picture of japanese knotweed! Sorry, I haven't a clue what this is - treating it with glyphosate, e.g. Roundup would be my suggestion. You could try it more concentrated than normal, as advised for treating knotweed.
Does it have a flower? The plant looks quite like Centaurea montana (which I've planted recently - gulp!) and it looks like a collection of buds at the top there

What colour are the flowers?

You could try this:
http://www.botanicalkeys.co.uk/flora/content/SEARCH.ASP

I have tried but can't tell from the photo what some of the answers would be.

It isn't Valerian, is it? (Centranthus in latin). The leaves look familiar - It has red, pink or white clusters of flowers, and is often seen growing in old walls.

Ooh, this is bugging me now! Could it indeed be valerian? I have never known it to grow under a metre of concrete, though it does grow in any nook it can.

No it is not valerian - I used to grow that. Knotweed crossed my mind but it is nowhere near as big. It does have a white flower. It grows up to about 12 inches but has a floppy habit and is not very self supporting. Just been round to the new neighbours and they have dug it all out but new shoots are coming up all over the place.

Looks like some kind of spurge - easier to tell when what passes for flowers opens up.

Just for info I checked my own valerian and the leaves are in pairs along the stem.

Could it be some kind of brassica, with those tightly-spaced little flowers?

Okay I have copied and pasted some answers from the link, if anyone could clarify or correct a few answers that would probably help:
What is the name of the plant you are interested in? = Don't know
Do you know what family the plant belongs too? = Don't know
What habitat does the plant grow in? = Don't know
What sort of a plant is it? = Herb
Where did you find the plant growing? = England
Do you think the plant might be a fern? = No
What month does the plant flower in? = Don't know
What colour is the flower? = White
What sort of flower or other kind of reproductive structure does the plant have? = Can anyone answer this?
How are the leaves arranged on the stem? = Alternate or spiralling along the stem
Are the leaves hairy? = Hairless
What sort of leaves does the plant have? = Simple
What are the margins of the leaf like? = Entire
Do the leaves have stipules? = Don't know
Do the leaves have petioles? = Think this is 'no'

Could it be yellow loosestrife, (lysimachia vulgaris) ? This spreads underground and can be invasive.

Another suggestion is some kind of horseradish, which the identifier came up with, it is apparently capable of sending its roots underground very far and coming up the other side.

If the main roots are woody and tough you may be better using a killer that you would normally use for ivy or tree stumps.

looking for a possible ID I was struck by how the definition of "invasive weed" varies. I've found websites listing Ailanthus, Buddleia, purple loostrife, various thistles & cornflowers, Elaeagnus, blackberry, ivy, you name it. I think it must depend on growing conditions and climate.

It isn't Lysimachia - the leaves are wrong, and mine is coming into flower now - the flower buds would be apparent on your 'specimen' and they don't seem to be.

Sorry Spritz,there are some which are invasive,I may have the wrong one :)

Thank you all for your ideas so far. I do not think we yet have the answer. Weed killer only kills the shoot cutting it off from the main root. Any small bit of root develops into a new plant. I live on the south coast and I have noticed it growing on some wasteland. I potted some up recently (this is the photo) and intend taking it in somewhere, Local garden centres or Wisley if necessary.

Do the roots look anything like this:
http://herbal-medicine.us/attachments/0000/0579/Armoracia_rusticana-3.jpg

Aster, I agree - some Lysimachias are out and out thugs! L. punctata is, for a start - the yellow one - but I still grow it because I like it! It gets heaved out occasionally to keep it under control....sort of...LOL.

I'm very puzzled about Goringfolly's mysterious plant - I'd like to know what it is, as well. Wisley's an excellent idea - they'll SURELY know!

Bryony has now solved this mystery. It is Lepidium Draba or Hoary Cress.
I reposted the problem with some more photos. Well done Bryony.

Yes, well done Bryony - I just recognise it as a weed and heave it out whenever I find it, never mind what its called, lol;-))

Thanks, Bryony - mystery solved! It doesn't sound like a nice plant to find in your garden!