Although not a game as such, I decided to include this here as it is such a good "getting to know you" activity for a group or class of children.
Give the children a thin, long rectangle of colourful paper each (cut these to an appropriate size for their age before hand) and put out a supply of crafty materials, crayons, glitter glue and so on. Ask the children to write their name in big letters on their strip of paper and decorate it any way they like.
You might like to ask any child who finishes very quickly to decorate a cover with the name of the class or group, or a title like "Our Name Book".
Leave the names to dry and then staple them all together inside the cover.