When I'm so exhausted I can't keep my eyes open. It could be at any hour- morning, noon, or night. I can only sleep in increments of two or three hours at a time, though once in awhile I'll surprise myself by staying asleep for a consecutive six hours.
Sometimes, even if I'm beyond exhaustion, I find that it takes an incredibly long amount of time to fall asleep. I'll be laying there in the dark thinking about something absurd. For example, last night I was dead tired, and went to bed around 3:00 in the morning, and I was just laying there with my hands folded on my stomach obsessing about beefing up my low level characters in FFXII and what a nuisance it was going to be. Next thing I know, I roll over and my clock says that It's 6:30 in the morning. Sometimes I'll just lay there, and see soldiers from Civilization IV shuffling about on a map, Tetris pieces aligning, or stupid cards flying around from Solitaire. And I can't get to sleep because these things are stuck in my head.
What's more annoying is falling asleep, and then waking up three hours later, wide awake... And knowing that in about six hours I have to drive somewhere.