The Cliflo API from New Zealand's National Institute of Water and Atmospheric Research (NIWA) gives access to data from 6,500 climate stations maintained by the government of New Zealand. The API webpage notes that access is by subscription, though the cost is free, suggesting that only registration and agreement with the terms and conditions are required. Access to data gathered by Pacific Island stations is restricted by agreement with Pacific Island countries. However it is possible to obtain access by permission.
If math is about measurement, then Mathletics may have your child's number. Much like games that post the best scores of players, the Mathletics SOAP API makes student scores accessible to programs as well as its website.
The Earthquake Data Portal API provides event information in both SOAP and REST-style services. For SOAP, the information is provided in QuakeML format. For REST, information is in either QuakeML or JSON formats. Users can do Event Searches, review Latest Events and look at Event Details. Data on seismic waveforms and their velocity are also provided, including the ability to predict arrival times and locations after they have been detected. The API will identify the data providers within the Seismolink network.