One of my fond memories of childhood was sitting in class, bored and surreptitiously playing Tic-Tac-Toe with the kid next to me. We passed the piece of paper back and forth, hoping the teacher would not notice and thereby learning absolutely nothing.
That was the fun of playing with paper. It could happen anywhere, it cost little, it was intergenerational and it provided an incredible amount of play power.
Using a pencil and paper to play a game was not the only enjoyment to have with paper. It could be folded into airplanes, footballs and cootie catchers (which some said could tell the future). Both involved careful paper folding and provided the thrill of either mystifying a friend with the cootie catcher or sailing a plane across the room while the teacher's back was turned.
Here are some of the games I remember:
Dots or Boxes
Hangman
Cootie Catcher
Paper Football Paper Airplane