My first thought is a nice way to save a stream in whatever format you want (e.g. transcode for watching on an old phone or something on your commute), just ffmpeg -i <stream> and then all your usual video format options, instead of having to download it and then convert it afterwards.
ffmpeg also has some processing abilities of its own, so you could e.g. greenscreen (chroma key) from a stream onto an existing video background.
ffmpeg is a pretty low-level building block and as others have said, it's mostly used as a library - a lot of video players or processing tools can now add support for stream inputs easily, and that's probably where the biggest impact is.
ffmpeg also has some processing abilities of its own, so you could e.g. greenscreen (chroma key) from a stream onto an existing video background.
ffmpeg is a pretty low-level building block and as others have said, it's mostly used as a library - a lot of video players or processing tools can now add support for stream inputs easily, and that's probably where the biggest impact is.