Why is it not possible to assign “maximize window” to key combinations? I can’t even record a macro to do that.
I can use alt-F5 to unmaximize a window (I’m not sure whether this setting is built in to LO), but I don’t have any way to assign “maximize,” or to record a macro to do it.
It appears it is possible. Why do you assume it isn’t? 
Resizing, hiding, showing and maximising windows is handled by the windowing management system, which is operating system dependent.
For Mac OSX, I can recommend the free and open source application Spectacle, but Ubuntu has that functionality built-in: press Alt+Space together, release and press M to maximise. For Windows, Microsoft lists keyboard shortcuts for Windows 7, including Windows logo key + Up Arrow to maximize the current window.