The family game night is probably the classic way to have fun for free. Have someone pick a game, and start playing. You can have the kids take turns choosing games or come up with your own way of choosing.
My husband and I do this sometimes. He’s a fan of Cheapass Games. They’re very creative, and yes, affordable. Very little resemblance at all to the games you’ll see in the stores. If they need standard game pieces, they just suggest you raid one of the probably many board games you already own.
With young children like mine you may have to be flexible on the rules. It’s not just about playing a game, after all. It’s about having fun.