What problem occurs by driving w/o replacing a bad catalytic converter?

What is your question?

with about 180,000, the light has come on a few times then gone out. Now it has stayed on for the last few hundred miles around town. I have brought into dealer who told me it was the converter though I didn't get the number. I'll go do that soon. In the meantime, what problems happen by failing to get it fixed before said problems develop? How risky is it in regards to causing more expensive problems?