Be like the Majestic Dolphin
---------

One option would be that animals would evolve to sleep like dolphin, half their brain at a time, though for different reasons. This opens the door to some interesting psychological and physiological features; 

 - For one, growth hormones are released during sleep, so this species may grow at half the rate of humans but for 24 hours a day. 
 - I imagine as individual sections of brain controlling movement rest, certain body parts may be paralyzed at certain parts of the day.
 - Depending on the sections of brain, they could possibly have different personalities, depending on what half was awake. 

When you get started here, you can really get rolling. I can imagine a species that instead of sleeping half a brain at a time; lives on quarters, or even eighths at a time. Keep in mind that if they evolved in this world, they wouldn't even have a concept of night; nevermind associate it with sleep.