Is there anyway to find out the history of the free app downloads? I know you could go to the market, however that doesn't update frequently, and am concerned my child is downloading unacceptable content and deleting it frequently.
I understand I am able to go to parental controls, however that isn't an option at this point.
I don't know of any way to capture that information after the fact. However, it is possible that the administrators of the market keep a record of downloads. Presuming that your child uses a device with the Android operating system, please launch the Android App Market on your child's phone, press the Menu hard key on the bottom of the handset, then select "Help". You should see a "Contact us with your questions" option. Select it and choose your preferred communication method.