Here's a list of a games that involve testing of one's strength (sometime the strength of a whole team, but sometimes each person on their own). These tend to be a bit more physical than many of the others, but certainly aren't dangerous. Well, at least less dangerous than most playground equipment
Note that these categories are pretty fluid and there is a lot of overlap between different ones. Where applicable I've cross-referenced similar games.
Games that also appear in other categories: