What is a good example of a conflict that occured in ancient times (think Romans, Greeks and Egyptians) that was a result of revenge?
It is hard to disentangle the various motives that lead to wars. This is especially true of wars that occurred in the distant past. However, one example of an ancient conflict that could conceivably be attributed to revenge is the series of wars between the Greeks and the Persians in the 5th century BCE.
These wars started because of Greek attacks on Persian outposts. The Persians ruled many Greek people who were living on the eastern coast of the Aegean in what is now Turkey. In the early 5th century, Greeks from the mainland of Greece attacked this area of the Persian Empire in support of rebellions by the Greeks who were under Persian rule. The Persian king, Darius, was enraged by this attack and wanted to take revenge on the Greeks. This was the beginning of the Greco-Persian Wars.