The fastest growing plant is Duckweed! It has got rid of half its genes and just focusses entirely on growth – very little immune system, no roots – just growth! Because it reproduces so quickly, it is becoming of interest as a new food source (well, to us, it is eaten in other countries already).
I don’t know if it’s the fastest of all, but bamboo grows really quickly, some species have recorded almost 1m in a day – that’s almost fast enough to see it growing before your eyes! This makes bamboo a really interesting plant for industry – there is a team of architects and engineers trying to build a skyscraper out of bamboo. It is also used to make paper and can be turned into a form of viscose that you can weave into clothes – in fact I’m wearing some bamboo boxer shorts today!