Tag was a game played by children. The object of this game was to chase other players and touch them, after which they would repeat the process with the other players.
In 2268, James T. Kirk referred to a transmission coming from an Orion vessel might be "someone out there playing tag". (TOS: "Journey to Babel")
In 2369, a de-aged Ro Laren and Guinan used playing tag as an excuse to affix combadges to the backs of Ferengi intruders on the USS Enterprise-D. (TNG: "Rascals")