Where do you want to start? When do you want to start? How far do you want to go?

For the history of ancient Mesopotamia, I recommend Roux, Georges. Ancient Iraq: George Roux. London: George Allen & Unwin, 1964.. Good overview, starts early, but restricted to the Mesopotamian region and the empires of Mesopotamia.

For a slightly wider (at least geographically) history of the Ancient Near East, I recommend Mieroop, Marc V. D. A History of the Ancient Near East: Ca. 3000-323 Bc. Oxford: Blackwell Publishing, 2007. This will touch on some of the other geographical regions.

This book touches more on the Roman period

Millar, Fergus. The Roman Near East, 31 BC-AD 337. Vol. 6. Harvard University Press, 1993.

And that's about where my expertise ends. I don't have a good book for you if you are looking for more modern history (history of Islam, modern conflicts, Israel-Arab relations, etc), that's outside my area of expertise. But if you want the beginnings and some ancient history, these books are where I started.

I'm here to suggest for you to read Isaac Asimov's The Near East, very simple narrative and could give you a wider idea on what you wish to focus on.

I think a good place to begin is Eugene Rogan's The Arabs: A History. It's not a complete history, notably leaving out the Persians, but it is one of the most comprehensive books on the region. The fact is that it will be hard to find a single book that will offer a complete history of the region. With that being the case, here is a link that offers many different books that you can start checking out to get a complete history of the Modern Middle East. As a quick sidenote, one book that isn't on that list that should be added is Colin Shindler's History of Modern Israel.