Akebia quinata, Russian vine, Clematis Montana, honeysuckle and Virginia creeper, google them to find out more ...good luck!

Depends how quick you want it......and the space to cover, russian vine was my first thought too but its not called "mile a minute" for nothing.....

annuals then you cant beat some of the nasturiums.

Seaburngirl is right, annual climbers have the fastest growth. Clematis takes 3 years to get into rapid growth as does Virginia Creeper, Russian vine etc. I actually find my Clematis Elizabeth is a faster grower than the Russian vine.

Nasturtiums can put on a few metres over a few weeks in summer with the right sun and water.

The question I'd ask is how big is the space you want this climber to occupy eventually - if its 40 feet by 40 feet, then Russian Vine or Clematis montana or Jasmine officinale will do the job admirably. If its 8 feet by 6 feet, you need something a lot smaller, and if its a shady spot, that limits your choice considerably.

Thanks for the comments, sorry been busy the last few days so haven't had chance to view the replies. Although I need a fast growing climber, I should have explained that it would be planted next to a small wooden arch (no bigger than a standard door way). I made the arch last year in a sunny patch of the garden and the climber died, as I'm impatient, I can't bare to look at the blank arch for another 12 months. You can see a picture of the arch on one of my photos.

Having looked at the pic of the arch, I'd go for clematis, one or two of the hybrids, maybe one up each side, one that flowers late and one that flowers earlier, so something like Clematis jackmannii or any of the hybrids in the late flowering group (Margaret Hunt, Perle d'Azur, etc) together with The President or William Kennet or Xerxes, which will flower earlier. In the meantime, for rapid cover, I'd grow Eccremocarpus scaber or Tropaleum peregrinum (easy from seed) to give cover and colour this year. You may need to regrow again for the following year if the clematis have not progressed much in the meantime.
Note also that the posts of the arch will need to have some kind of supports attached - the easiest is probably clematis mesh, so that the plants can attach themselves to that and climb upwards. Even twining plants like honeysuckle will need something to support them, because the posts of the arch are too thick for them to twine around.

That's great bamboo, thanks very much. I'll be taking a trip to the garden centre over the weekend so I will keep an eye out. I will also add the mesh, didn't realise that climbers needed the added support so once again, thanks for your advise.