What is the yellow greenish smelly ball people cough up?

The little white particles that you are coughing up are called tonsilloliths and except for their bad smell are not a condition to be worried about. Tonsilloliths form when mucus, bacteria, and debris condense into small particles on the surface of the tonsils. These odorous balls of material are sometimes coughed up. Having tonsilloliths does not automatically mean that your breath is offensive as tonsilloliths contribute very little to bad breath.

Additional information - I get these 2-3 times a year now for 22 years. My "chunks" are slightly larger than a pea and seem to come from deep in the chest after an extensive coughing spell (with awful breath and a low grade fever). One I coughed up was an off white color and unusually large so I chose to section it on a piece of glass.

I found a core in the middle of a soft white folded coating. The odor was absolutely unbearable cutting into the chunk. My thought was perhaps I was inhaling small pieces of nuts into my lungs where the body formed the mass to get them out by coughing so I quit eating nuts but they persisted.

You can definitely tell from the awful smell that one is "growing" and that smell goes away after it has been expelled, often at high velocity

Years ago, I asked my MD about them and she gets them too but had no idea what they were. Its nice to know I'm not the only one with this and it has a name.