Contract period: Automatically extended for three year periods

The plural “periods” tells you that at the end of each period, it will be extended for another period, and so on indefinitely.

If they meant only one additional period, they would have used the singular “period”.

In both cases, you would expect to see some additional language about how one or both parties can prevent the automatic extension; usually in such contracts neither party can cancel a period that has already started—at least without cause—so this is the only way to keep it from becoming permanent.