Perhaps you've heard the theory that Moses and Akhenaton were one and the same.  That's hard to believe.  If the man was a king, why would legend make him a prince?  More likely, Moses was Egyptian royalty.  Why would legend make him an adopted prince?  Simply because some of the Biblical authors were intent upon preserving the myth of ethnic purity and therefore Moses could not be tied to the Egyptian royal bloodline.

We cannot say for sure who Moses was.  We can make an educated guess based upon the Book of Exodus.  If the daughter of Pharaoh was the actual mother of Moses, then Moses was the Grandson of Pharaoh.  That Pharaoh was the same Pharaoh who did not know Joseph and enslaved the Hebrews.  That Pharaoh was Horemheb.

As far as we know, Horemheb did not have any children with his principle wife, Queen Mudnodjme.  However, it was common for Pharaohs to have concubines.  Furthermore, before the reign of Ramses II the Pharaoh's harem children were not mentioned in the hieroglyphs. (Horemheb was about 20 years prior to Ramses II.)

Here's a scenario of what may have happened.  The wet nurse of Moses had a son named Aaron who was a high priest of the outlawed Amarna religion.  Aaron made Moses a believer, but when Grandpa Horemheb found out, he exiled Moses from Egypt.  Moses went to Midian, an area now in northeast Arabia along the Red Sea.

When Horemheb died and left no heir, Aaron convinced Moses to return to Egypt and claim the throne.  Unfortunately, by the time Moses got back, Horemheb's Vizier consolidated power as Pharaoh Ramses I.  Moses knew he had no chance in Egypt.  He turned around to return to Midian, but Ramses still feared Moses as a potential rival and sent a military expedition to kill him.  The military expedition met up with disaster.  Moses arrived safely in Midian and lived there the rest of his life.

The exiled Amarna followers told the story of the escape for generations, but it was not in Midian where Amarna refugees established themselves.  That happened in the remote hill country of Canaan.