The longest neurone in the human body is one of the peripheral neurones in the leg, travelling from the base of the spine to the foot - over 1 metre in some people!
Running this a close second is the neurone from the motor cortex to the lower part of the spine...

These are single cells, with highly specialised axons that span huge distances for their size. The question now is: how long could they get? If they grow proportional to a person's height, are they capable of indefinite growth if given sufficient nutrients?

Over a metre long and not visible to the naked eye...

(http://diamidinophenylindolee.tumblr.com/post/59584066686/single-neuron-under-the-electron-microscope)