You can make play dough without cream of tarter, but it won't stay fresh as long as it will when cream of tartar is included.
Here's my favorite play dough recipe and it's from the book, Don't Move the Muffin Tins, by Bev Bos :)
2 cups flour
1 cup salt
1 teaspoon cream of tartar
2 tablespoons vegetable oil
1 teaspoon food coloring
2 cups water
Mix ingredients in a saucepan, over medium heat, stirring constantly, until the dough leaves the sides of the pan.
Remove from pan and knead for a few minutes.
Store in a tightly closed plastic container or a zip-lock bag.