In response to question edit:
Without any free space of any sort, the system is quickly unusable. You can't save work in any application, several applications will fail because they back calculations with the disk, etc. You can't even reliably use only a web browser, because you'd not even be able to patch it in the event bugs are discovered (which they invariably are).
Very few computer tasks are going to succeed/make sense if nobody can save anything ;)
By "On the system partition" I'm assuming you have some other partition which does have free space. Any system without any free disk space at all quickly becomes unusable.
Directly, the worst thing that can happen by having a full system disk is failure of the system to save minidumps upon blue screens of death.
Indirectly, however, there is a bigger problem. Several applications, when they install themselves, will attempt to install DLLs into %WINDIR%\System32, even though they really shouldn't be putting anything there. Any program which does this will fail to install, because (of course), the disk upon which that folder rests is full.
This also makes doing things like Windows Update, or for that matter, any other kind of update or installation, because they often try to either write to the system drive, or at least to use the system drive as a temporary location.
So, while Windows itself shouldn't generally fail as a result of a full system disk, there are plenty of applications which would fail. It's therefore probably not a good idea.