Two very simple games that require only pencil and paper.

Noughts and crosses is a very simple game that involves drawing a grid of two horizontal lines and two vertical lines. First person to go decides if they are a nought or a cross. They draw their symbol in one of the 9 squares created by the grid of lines. The second player takes the other symbol and places that in an empty square. The aim of the game is to get three symbols in a line. Play alternates until the grid is filled up or there's a winner.
Small children can enjoy playing this game.

Older children will enjoy hangman. The first player picks a word and draws a line space for each letter of the world. The second player has to guess the word. Each letter guess results in either that letter being inserted into the spaces in the word or a hangman gets drawn. Rules vary but most children will enjoy playing more if letters they have picked are written down when they are picked so they can see which they have already guessed at. You can change how many lines make up the hangman to make it easier for the child, or invent a new shape you can draw with simple lines.