Will one be refused entry on a full passport?

What happens at a border when you need an additional stamp, but don’t
have empty pages, or even no free space left?

This depends on the entry procedures at the border.

At almost all border posts, your passport has to be stamped; this stamp is later verified at multiple check points. This stamp is your proof of legal entry into the country or territory. This stamp is required even if you hold valid travel authorization to the destination country; hence most ports will deny you entry if your passport is full.

However, certain countries have “e-gate” procedures. This means that you can cross the border without carrying a passport. You instead travel on your country ID, and biometric data is scanned to log your entry. Obviously here you don’t need a passport at all, so the free page requirement is also moot.

Even with e-gate enabled ports, it is always recommended to carry your passport and make sure that it has free pages. This is to ensure that should the e-gate be under maintenance, you are still able to cross immigration.

The most popular e-gate enabled border post that I have been to is Dubai International Airport (OMDB); where there are dedicated e-gate lanes that are completely automated (no immigration officer at all).

In addition, GCC (Gulf Cooperation Council) citizens can travel across GCC member states if they have a digital identification card. In this scenario, a passport is not required.

Some countries will offer you a separate “stamp book” if you travel often. This book is to make sure you don’t fill up your passport during frequent transits. I have used such a book when residing in Saudi Arabia and studying in Bahrain. You need both the book and the passport. The passport has the visa, but the immigration officer will stamp the book instead.

Some countries offer their citizens passports with extra pages at nominal costs; you can request this at the time of passport renewal.

I have an anecdotal answer to this question, as I once gave shelter to a British “travelling bum” who was hitchhiking from Australia back to the UK in the 70s (save for the sea voyage). The chap found himself penniless and destitute by the time he reached India, but miraculously came across some tourists in a car who were willing to give him a ride all the way back to the UK. The problem occurred at the Pakistani border, when the border official wouldn’t allow him entry since his passport had absolutely no space left for a stamp. He rushed to the British Consulate while his ride waited, but he didn’t have the few British pounds required for new passport pages and/or a new passport. He broke down, figuring that his fate was to die a silent, anonymous death in an Indian slum. Fortunately, British officials relented and granted him an emergency loan to resolve the passport issue, and even dispatched a Consulate limousine to reunite him with his ride.

Often, the laws are structured in such a way that you very much want the entry stamp or visa, for example by putting the burden of proving you entered legally and did not overstay on you, the visitor. Assuming you somehow manage to enter, it will be more difficult to prove when and how you did it so you could for example be considered as a an overstayer with all the negative consequences (fine, etc.).

I am not sure what would happen if you show up at the border of a country like that with absolutely no space in your passport but entering is only the start of your problems. You still need to leave the country and/or interact with the local authorities somewhere down the line.

From personal experience, I can tell you that I once (almost) was not allowed to board a plane because, at checkin, I was told I needed at least two empty pages in my passport to allow for stamps and visas. This was (and is) a requirement of the destination country, even though it had little relevance to me (I already had a full page visa for said country, while the stamps are small).
It took a lot of grovelling on my part to be let on board.

It stands to reason that if you can’t be stamped in, you won’t be let in. I presume that this is why plenty of countries require two empty pages in your passport, upon entry: One to stamp you in, one to stamp you out.