Here are some of the ideas on what to do with the tops of the T-shirts.
How about using the neck ribbing for baby bibs or adult bibs made with towels that slip over the head.
Knitting Rose's idea is:
You can sew up the neck, arm holes & bottom - fill with rice and use as microwavable heating pads. We do this with our old t-shirts (I use the bottoms of them for rags). They fit across the back of your neck and shoulders and hold the heat well.
stuff them and making pillows or quilts
Marilynart's idea is:
I have slipped this sort of thing over a coat hanger to secure dresses with spaghetti straps.