The civilization that comes to mind is the Hebrews, or Israel; however, they weren't all that large. The impact they had is quite massive though.

If we look at just the Bible, Egypt, Mesopotamia, and Phoencia all play a role in Hebrew civilization, as does Persia and Greece.

Starting with Egypt. A massive portion of the early history of Israel deals with Egypt. The most clear example of this is with the story of Moses. William Dever, in Who were the Ancient Israelites and Where did they Come From, while dismissing the Exodus story as a whole (as most scholars do), does suggest that some of those who made up later Israel in fact came from Egypt and helped create the national legend.

We also see this same sort of thing with Mesopotamia, specifically with Abraham who is said to come from that area. We also see the Garden of Eden being placed in Mesopotamia. While scholars such as Israel Finkelstein in his popular book, the Bible Unearthed (as well the Quest for the Historical Israel and elsewhere), again nearly unanimously agree that the stories most likely are not historically accurate (and the archeological record doesn't support the history laid out), there is most likely some Mesopotamian influence that helped make up Israel.

With Phoenicia, we have in the book of Ezekiel (and elsewhere), mention of different Phoenician towns. The one that comes to mind is Tyre, a city-state that Ezekiel tells us was once closely united with Israel, but later on distances themselves.

The History of Ancient Israel, by Michael Grant also goes over the conquests of Israel by Persia, then Greece quite well. He pieces them together and shows how both cultures helped transform Israel.

And while Israel wasn't a very large country, it did have a massive impact. It is from Israel that we get the Abrahamic faiths, that have had a profound impact on our history.