The ISS Notify API (or is it the Open Notify API?) was written by Nathan Bergey for a Science Hack Day competition, then released to the public. You can use it to find the location of the ISS, or to find when it'll pass over a location! Check it out.
The Space Telescope Science Institute works closely with the Hubble telescope and its operation. They've made available, through their HubbleSite API, tons of photos, videos, and other data.
NASA's API makes their data (such as Mars rover photos) available to anyone who wants to consume it. It's an unprecedented wealth of knowledge, so let's dig in!