The most common need for safety testing is if you plan to sell your books or games to a young demographic through a retailer. You will need to make sure your product meets safety requirements designated by each country.
The most common safety tests are done to verify that a printed product meets all requirements of the CPSIA (Consumer Product Safety Improvement Act) in the USA.
However, for games, additional testing may be done to verify that a game is suitable for specific age groups (without testing, games can only be labeled for ages 13+). Games that will be sold in Europe may also want testing to verify that their game meets CE marking requirements.