I've had the pleasure of reading Gombrich - The Story of Art; Illustrated, McPherson - Battle Cry of Freedom; Illustrated, and I'm about half way through Davies - Europe: A History. What am I missing?
If you haven't already you might have an interest in taking a look at our /r/askhistorians booklist: https://www.reddit.com/r/AskHistorians/wiki/books
As for "What am I missing," well, given my area of expertise, I would say you're missing quite a lot.
As mentioned below, one-volume history texts aren't common in academic history but there there are two really important works of synthesis I would recommend:
Fernand Braudel's highly influential The Mediterranean and the Mediterranean World in the Age of Philip II. It's a vast book that uses the Mediterranean sea as it's point of departure and examines everything from geography, to institutions, to war, and agriculture.
J.H. Parry's The Spanish Seaborne Empire. Another classic overview, it examines the Spanish conquest and the development of Latin America.
Frank McLynn. "Genghis Khan. His Conquests, his Empire, and his Legacy"
While it doesn't go into great depth about the specific Khanates like the Ilkhanate, Yuan Dynasty, Golden Horde, and Chagatai Khanate, it does provide good information on what set the stage for the conquests and how they played out, as well as some details on Mongols society, culture and tactics.