Initially, I held the position that sleep was probably more important than eating food, and that you would die from sleep deprivation before you died from starvation. We looked it up, and learned about a disease called fatal familial insomnia. A mutated protein causes the onset of permanent insomnia, and there is no cure. It affects about 100 individuals worldwide. [1]
The average life expectancy of a patient is 18 months after the onset of symptoms. The first 9 months is a worsening case of insomnia, where the patient will experience paranoia, panic attacks, and hallucinations. Sleeping pills and barbiturates have been shown to worsen the clinical manifestations and hasten the onset of the disease. The individual then becomes completely unable to sleep, and will enter a state of dementia before becoming completely mute and unresponsive. This second state of permanent insomnia lasts for 9 months.
So ultimately, we learned that you can go for 9 months without sleeping before you die. I'm not sure at what point you'll suffer irreversible brain damage, but that was an interesting finding for me nonetheless.