Mine do not stay on either. After either timing out by itself or after I touch the on-off button to turn off the screen, it stays off all night. I only know charging is complete when I wake up the screen.
Found this over on xda. Sounds like it has some problems though. This battery full message is super annoying. I use my phone as a clock at night so my screen is on and the popup lights up my whole room.
Stuff like this (how could a programmer NOT be aware that most folks plug their phone in to charge before going to sleep?) makes me wonder if Samsung, or Google, hired grade school kids at summer programming camp to write the software environment on these phones.
I'd call my Treo "ancient and venerable" if it wasn't staring at me and gloating from the corner.