Its not possible. Eventually, the brain will make itself shut-down, and you would fall asleep. Its a protective feature within the brain. You would notice yourself getting light-headed, drowsy, dizzy, etc. Eventually, you would fall asleep to allow the brain mechanisms to take over and restore to balance neurochemicals.
There was a guy who wanted to fly a hot-air balloon around the world, or something like this. He attempted to do so without sleep. After being awake for 20-some days, he eventually fell asleep, baloon descended, and he was unsuccessful in his venture.