Personally, my 5 recommendations would be:
1). The Black Middle by Matthew Restall -- It kind of takes Richard White's "middle ground" model and applies it to the position of people of African ancestry living between the Spaniards and the Native Americans in the early modern Yucatan peninsula. Cool look at an historically overlooked group of people and the way they crossed socio-economic strata and mediated between the conquerors and the conquered.
2). The Return of Martin Guerre by Natalie Zemon Davis -- This has had its share of criticism, as Davis fills in a lot of the lacunae with her own speculations. However, it's a thrilling read, and it raises important questions about how much speculation and possibility historians can/should include alongside more "concrete" facts.
3). Comforts of Home by Luise White -- This is an examination of the prostitution industry in colonial Nairobi, Kenya. The range of sources it synthesizes (everything from oral interviews to government records to private organization records) is impressive. I like the way White avoids falling into either the "prostitutes were simply exploited and had no agency" trap or the "prostitutes were inspiring, empowered women and had total agency" trap. She presents a very nuanced, comprehensive picture.
4). Changes in the Land by William Cronon -- This is one of the most important books in the canon of environmental history. Cronon is very persuasive in his arguments against ecological equilibrium theory (the notion that nature without humans is basically stable), and shows how other animals, plants, geology, and climate exercise their own agency in a way. Cronon also does a great job of breaking apart the "ecological angel" (or "noble savage") trope, which posits that Native Americans lived in perfect harmony with nature and didn't alter their environment. Cronon shows how the Americas (pre-Columbus) were very inhabited and were very much modified by the inhabitants.
5). The Making of the English Working Class by E.P. Thompson -- Even despite having a Marxist bent (and I am not a Marxist or leftist), I like this book because of the way Thompson pioneers the analysis and importance of "experience" as an historical phenomenon. Instead of trying to provide a dry, "fact"-based, statistical narrative about how a working class originated in England, he draws on a rich mixture of sources to try to get at how English laborers experienced their situation mentally, emotionally, and physically. Thompson shows how these experiences made English laborers feel like they were part of a "working class."
Matthew Restall Seven Myths of the Spanish Conquest is a mind-blowing book. He establishes seven persistent myths of the conquest, then breaks those myths down in one brief volume. Forget what you think you know about the early colonial period, and be prepared for a deeper, richer story than you could ever imagine.
Daniel Richter Facing East from Indian Country: A Native History of Early America is a great introduction to eastern North American history. The big appeal of this book is shifting the narrative of contact away from the European perspective, and instead anchoring the story in Indian Country. A great book to challenge how you view contact.
Andrés Reséndez The Other Slavery: The Uncovered Story of Indian Enslavement in America is the single best introduction to understand the temporal, geographic, and cultural magnitude of the native slave trade in the Spanish Empire. Absolutely vital for understanding the history of the Americas.
Colin Calloway One Vast Winter Count: The Native American West Before Lewis and Clark is the best introduction and overview of the American West. I absolutely adore this book.
I'll also echo the previous mentions for Cronon's Changes in the Land, because no other book really addresses the fundamental ecological impacts of colonialism, and how those changes prohibited the continuation of previously established Native American land use patterns.
Let’s narrow it down to Chinese history because I can, oh and also specifically 1800-1911.
Discovering History in China by Paul A. Cohen (1984): An absolutely indispensable work on the American historiography of China up to the 1980s. Cohen explains the origins of and flaws inherent in the schools of thought which dominated until the mid-70s - ‘impact-response’, ‘modernisation’ and ‘imperialism’, and observes the emergence of a ‘China-centred’ school which eschews the Eurocentrism of its predecessors.
The Search for Modern China by Jonathan D. Spence (3rd ed. 2010): Probably the best general overview of modern Chinese history there is, starting with the Qing conquest and culminating in the then-present day. Spence masterfully combines his own expertise as a cultural and intellectual historian of China with the political narrative for an excellent big-picture view of the last 400 years of Chinese history.
Collapse of the Heavenly Dynasty a.k.a. The Qing Empire and the Opium War by Mao Haijian (1996, English translation 2016): The absolute most comprehensive account there is of the First Opium War from the Chinese perspective, examining the conduct of the Qing government instead of simply praising Chinese and excoriating British actions in the style of the traditional narrative established in the 1910s. Both a comprehensive work of history and an interesting commentary on the nature of autocracy.
The Opium War by Julia Lovell (2011): I was unsure about this one as it derives so much of its narrative of the war from Mao, but ultimately included for treading such different ground in other respects. Lovell looks not just at the political side of the war and the Qing but also the cultural, highlighting the cultural divides within China exposed by the war and the long-term impact of the war, of imperialism and of opium on Sino-Western cultural perceptions.
Autumn in the Heavenly Kingdom by Stephen R. Platt (2012): It is easy to think of the Taiping as ‘hurr durr worshipping the brother of Jesus’, and thereby obscure their real and significant political and cultural impact. Platt places the Taiping firmly in a geopolitical and domestic political context, investigating the reasons for Western military intervention, the nature of Taiping reform proposals and the character of the loyalist general and administrator Zeng Guofan. I’d also have recommended his Imperial Twilight but I already have two Opium War books on this list.
Also a couple of honourable mentions:
Hmmm. I think the way I would answer this is in terms of five history books that have personally had the biggest influence on me. These are all except for one books I read in my undergraduate studies:
A Short History of Reconstruction by Eric Foner - I've read the entire work, but you don't necessarily have to! This is the abridged version I read in college, and it was a big challenge to a lot of preconceived notions I had about American history, down to the fact that even the use of terms like "carpetbaggers" and "scalawags" in discussing the Reconstruction South is the adoption of a political narrative that was victorious after a widespread campaign of insurgency and terrorism.
Ecological Imperialism by Alfred Crosby - From the historian who brought you "the Columbian Exchange". Possibly the definitive account of how European expansion over the past millennium involved changing the very plant and animal species in areas of encounter, and how this was often intentional.
The Crusades Through Arab Eyes by Amin Maalouf - An historical essay on the Crusades, drawing on Arab primary sources of the time. I'm not necessarily sure how much it holds up history-wise, but I found it an exceptionally interesting history that takes the story away from the Crusaders, and instead tells it from the perspective of Middle Easterners of various religions and languages who fight and argue among themselves and also try to make sense of these invading barbarians.
Hitler's Army by Omar Bartov - Especially appropriate for this subreddit, this was probably the book that broke the "good Wehrmacht" myth for me, and was therefore a hard piece of work to grapple with. It's in the vein of Browning's Ordinary Men but came before it, and changed a lot of the tone in discussing Germans in the war, how much they knew and talked about atrocities on the Eastern Front, and how willing they were to be complicit in them.
Reconstruction of Nations by Timothy Snyder - Snyder has both his political and academic detractors, but this is I think one of his better (and earlier) works, and really helped me to grapple with how fluid national identities could be in Eastern Europe - again, with an Eastern European background and being steeped in particular national narratives, this punctured a lot of specific ones I held. It covers the development of national identities in Poland, Lithuania and Ukraine in the late 19th and 20th centuries, and how these different national identities interacted with each other, often with tragic results. It stands apart from his other works in that it is not about Germans/Nazis and Russians/Soviets, except insofar as how they were strategically used by competing nationalists in their struggles with each other.
It's more a political science essay but I think I'll give an honorable mention to Max Weber's The Protestant Ethic and the Spirit of Capitalism, in part because it's an interesting counterpoint to the broadly Marxian idea of economics driving culture (sometimes the reverse happens too!).
Mine are going to bend towards Postwar Europe and the United States, so climb aboard! Several are not University Press, but I greatly enjoyed them.
1-Burned Bridge by Edith Sheffer: it's highly specific to postwar Germany, but helps to show how the Cold War division of Germany shaped the development of two towns that had previously "co-developed" in many ways. It's an interesting process to see them split.
2-Ordinary Men: Reserve Police Batallion 101 and the Final Solution in Poland by Christopher Browning. Some academic reviewers take issue with his sourcing (using legal testimonies because if a lack of diary/day-to-day accounts) but I find it a strong account of how more than 80% of this Batallion willingly participated quite actively in the Holocaust. People like to ask "why?" and this book gives an answer (though hardly THE answer, nuance abounds).
3-Case Closed: Holocaust Survivors in Postwar America by Beth Cohen. I stumbled across this book while casting about for an undergraduate thesis topic, and loved it. In essence, it charts (through exhaustive archival research) the lives of Holocaust refugees who moved to America. Despite many reports on their rapid integration, the truth is mixed. Aid agencies rushed to "close cases" and get refugees off their rolls and into society to fit in the narrative of easy success. This book charts the reality they faced.
4-The Good War by Studs Terkel: a classic, through Oral Histories, we see how World War II impacted many lives of those who lived it. I read it in the context of studying how the US Military interacts with American Society coupled with the "challenge" of masculinity when men go to war, and found it very interesting from that context. Tim O'Brian's "The Things they Carried" is a bit similar (though very different narrative styles) and addresses Vietnam. It is interesting when viewed through that lens as well.
5-for a random outlier in my list, I greatly enjoyed "The Future is History: How Totalitarianism Reclaimed Russia" by Masha Gessen. Not academic, and arguably a bit more political and certainly more charged, but something I found very intriguing about how Russia has developed since 1991. It tells the story of a number if Russians who saw the hopeful rise of democratic norms, and charts the collapse of those norms. I'm not an expert on Russia, but I found it well done and was recommended to me by an undergraduate Russian historian in my department.