The Dog API provides a random dog image from a dataset of over 20,000 dog images. Developers can integrate a completely random image or specify a breed or sub-breed. The JSON API can also return an array of all the sub-breeds from a given breed. This API is free to use, but their website includes ads and there is an option to donate to the developers.
The Bandsintown API provides read-only data that includes information about a given artist as well as events related to that artist. Developers can call the artist data and provide an artist, and the API will return a link to the bandsintown page for the artist, a thumbnail image, an actual image, and a Facebook Page URL. Additionally, they can ask for event data and the API will return an array of events, venues, and more within a date range. To use the API, developers must contact Bandsintown with what they plan to use the API for and request an app ID. It appears to be free to use, but might depend on the developer's intention with the API.
The Open UV Index API provides real-time global UV index data. Developers provide latitude and longitude coordinates, and the API responds with the UV index and time it was recorded, UV max and time, as well as the ozone level and time. The API is free to use for personal users who make under 50 requests per day. However, for more rigorous use, it'll cost $15 a month or more, depending on how many requests will be made.
There are countless JSON APIs available to integrate on a website. Some are free, some come with a cost, but any of them can provide valuable data and resources for developing with JavaScript.