Excluding /Library (sic, not ~/Library) from Time Machine
Apple and third-party applications may have a variety of subsidiary files stored in /Library.
Much of it will be in /Library/Application Support.

Third-party fonts may be in the Fonts subfolder. 

But there may also be Dictionaries, Color Profiles, LaunchAgents, Privileged Helper Tools, system-wide Preferences (including network settings), printer drivers, QuickLook and Spotlight plug-ins, AppleScript Scripting Additions, Frameworks, and so much more, depending on what you've installed. Anything for which there's a named folder!

My advice is that you probably won't save much space on a backup disk by excluding /Library, and if something goes wrong, it's easier to restore one file or one folder than have to re-install a whole load of applications and the OS just to be sure.