There's a lot of (sometimes contradictory) information to be found around the internet about a Carnet de Passage. The linked Wikipedia article, for instance says that
a Carnet de Passage is required in Burundi, Egypt, Kenya, Libya, Senegal (for vehicles older than 5 years), Southern African Customs Union (BW/NA/LS/SZ/ZA), Sudan, Uganda.
Well, last year, I've been doing some overland traveling with a friend and we drove a 15 year old car without Carnet de Passage (besides through Europe) through Syria, Jordan, Egypt, Sudan and Ethiopia. And I can tell for sure that the only country which was kind of problematic (but not even impossible) to enter without a Carnet was Egypt.
We had to actually buy a local Carnet which was only valid in Egypt and costed us around 500€. All in all it was much cheaper for us to travel without a Carnet as the cash bond of 5000€ (which is the minimum in Europe) was much higher than all the money we spend on the car, including the purchase and all temporary import taxes we had to pay at the different borders (we knew in advance we wouldn't bring the car back home).
From other overlanders we met, we heard that the only countries in the world that are kind of "problematic" concerning a Carnet de Passage are Egypt and Iran.
I've been thoroughly searching for this info before we left, and I could only conclude that the only possibility to know for sure was to just show up at the border and hope for the best.
With this question I would like to gather real-life, hands-on experiences with overlanding without a Carnet de Passage, with the ultimate intent to create a trustworthy reference so future overlanders don't have to hunt around the internet to only find contradictory information and in the end buy themselves a Carnet de Passage while it wasn't really necessary.