Western cultures, to the best of my understanding, can trace their roots back to Ancient Egypt. Many modern day Western cultures took influence from Rome, which in turn took influence from Greece, which in turn took influence from Egypt, but did Ancient Egypt (3100 BCE to roughly 400 BCE) take influence from another as well?
Thank you to those that take the time to answer!
When you're talking about Ancient Egypt, that's a ridiculously long period of time. Also, like real roots, there are usually many cultural roots rather than a single one, and neighbouring cultures tend to cross-fertilise. That's a lot of roots! If you want to start, just look at Ancient Egypt's powerful neighbours, almost each one has influenced it in some way. So I'll try to categorise them instead of listing every single one.
Upper and Lower Egypt
Ancient Egypt is the product of the merger of two very ancient kingdoms: Upper and Lower Egypt. Before this there were separate towns. So the two biggest roots of Ancient Egypt are Upper and Lower Egypt. Perhaps the biggest cultural influence is in their mythology: Horus, associated with Upper Egypt, is said to have conquered Set, associated with Lower Egypt.
Mesopotamia
Mesopotamia, the Cradle of Civilisation and very close to Egypt, is obviously a big influence:
A new and distinctive pottery, which was related to the pottery in the Southern Levant, appeared during [the Early Dynastic Period]. The Mesopotamian process of sun-dried bricks, and architectural building principles—including the use of the arch and recessed walls for decorative effect—became popular during this time.
Another example is the chariot, which passed into Egypt from Asia.
The Egyptian chariot betrayed its Asiatic origin in a number of ways, by the names of its parts which were Semitic and by its decorations which often took the form of date palm branches or animals opposing each other, both Syrian motifs.
http://www.reshafim.org.il/ad/egypt/timelines/topics/chariot.htm
Nubia
The region south of Ancient Egypt interacted with it in various ways, conquering and trading with each other. Nubia even contributed some pharaohs, who (along with Nubia itself, eventually) assimilated with Egypt because it was the "greater" culture. But there was a lot of Nubian influence:
As expected, strong Nubian features and dark coloring are seen in their sculpture and relief work.
Ancient Libya
Very little is known about lands west of Ancient Egypt, but they did at least contribute some pharaohs, who brought some Libyan practices to Egypt, at least for a time.
Ancient Greece
Even though you most often hear the cultural exchange being one-sided, Egypt to Greece, and Herodotus described the Ancient Egyptians as xenophobic, but there were large Greek settlements in Egypt, so there is probably a little bit of Greek influence. For example, there are some parallels between Homeric epics and the contemporary tales of Setna Khaemweset, although this is pretty weak evidence.
Obviously Egypt was later Hellenised after Alexander's conquest, but that also marks the end of Ancient Egypt.
Greece influenced Egypt heavily. After the conquests of Alexander, Ptolemy was installed as the rule of Egypt, and started the Ptolemaic dynasty. Through the centuries, we can see a big interchange between Greece and Egypt. In fact, the rulers of Egypt for centuries were white, Greek-speaking Europeans. Eventually, Egypt was brought into the fold of the Roman Republic, and later Empire. Some culture came out of there through inevitable interchange, but AFAIK, the majority of the influence was exerted upon them in requiring people to speak Greek to take part in gubernatorial business. Your layman might retain more "Egyptian-ness" than a member of the aristocracy, but for the most part, Egypt was acted upon by its conquerors. Alexandria, for example, was a fairly Greco-Roman city.
There was of course military culture exchanged. Numidian cavalry was considered quite powerful, and influenced military tactics in the Italian Peninsula and greater Europe.
Though it is outside of my realm of expertise, AFAIK, the client states such as Numidia maintained more of their own culture than the Greco-Romanized Alexandrians.
The following occured much before the greeks: I think a good point to take into account are the burial customs which evolved and reached us to the modern times as The book of the dead. The translation given by Wallace Budge gives in the introduction an account of how such funerary rites (and religious beliefs!) influenced local people that later was called Egyptians. If I remember correctly, the immigrants came from Asia, but the text never says explicitly from which civilizations they were!
There have been a couple of good answers mentioning various outside influences. I chose to answer this question because two other important influences are not yet mentioned. Notably the Proto Canaanite translation of some parts of the hieroglyphic text in the Pyramid text of the Pyramid of Unas. That means Proto Canaanite utterances were used in the Pyramid text. And the first pyramid site Saqqara was named after the Mesopotamian Ziggurats that influenced the Ancient and contemporary landscape considerably.
Grain was imported into Egypt. It was cultivated in the highlands of what is now modern Turkey. And grain had a significant influence on Egypt and the world because it lies at the foundation of civilizations, because so many people could be fed ’suddenly’.
Some more reading;
A link for more info on the Proto Canaanite parts of the Pyramid Text:
"Earliest Semitic Text Revealed In Egyptian Pyramid Inscription", ScienceDaily, January 30, 2007
On pyramids:
Joyce Filer: "Pyramids", Oxford University Press, 2006, p44.
Contacts with Uruk https://en.m.wikipedia.org/wiki/Foreign_contacts_of_ancient_Egypt
