Chamber Potty
Of course, chamber pots were the main reason every palace was so nasty all the time. Modern plumbing was several decades away, so this was the next best thing. Even the kings and queens of the era were subjected to this kind of bathroom. While it certainly wasn't ideal, it was better than the alternative.
Into the Streets
Once the chamber pot was filled, there was no flushing mechanism to wash it all away. Instead, people would just kind of leave it anywhere. Not only that, but they wouldn't confine themselves purely to the toilet either. People relieved themselves in hallways, staircases, even on the street.