Primary annalistic sources for ancient Mesopotamia and Egypt

As an Egyptologist, I can only answer about Egypt. In my answer, I will focus on annalistic material issued directly as such through the Pharaohs and their circle (royal context).

Despite the fact that early detailed records do not survive, we can say that early Egyptian royal catalogues functioned, in a way, as non-detailed annalistic records. For example, in the (fragmentary) Palermo Stone (2392–2283 BC), the earliest list of Pharaohs in Egypt, significant events in the reign of each Pharaoh are included within each "entry". Another, more detailed, piece of early Royal Annals is the so-called "genut" of Mit Rahina (Memphis), dating to the reign of Amenemhat II (The Middle Kingdom, 12th Dynasty, c. 1929–1895 BC). During the New Kingdom, the Pharaohs and their circle produce more detailed records on their deeds, destined to be placed upon visible spaces if temples and other monumental constructions, as Egypt becomes more outgoing with regards to warlike activities as well as diplomacy. Thutmose III's annals, describing his wars in Palestine, Ramesses II's Kadesh and other war-related inscriptions on various monuments and Ramesses III's Sea Peoples related texts in his funerary temple at Medinet Habu are the more prominent examples of such records, although there are much more.

With regard to Harkhuf and other autobiographical material, as well as material from Amarna, we can say that although it can function as a group of historical sources, it is not annalistic per se, as it had not been composed as such. As for Sinuhe, it may contain historical info but it is pure literature. Still, it is worth mentioning those as, in a way, they reflect a tendency of the Egyptians to move from less to more detailed sets of records adopting narrative to record historical events.

For more see

Wilkinson, Toby A. H. (2000) Royal Annals of Ancient Egypt p.23ff. New York: Columbia University Press.

The Date of the War Scenes at Karnak and the History of the Late Nineteenth Dynasty, http://eemaa.org.gr/bibliography/1072

I have to admit this will not be an entirely satisfactory answer and I hope someone else can contribute a fuller one. However, I have taken several evening or weekend classes in Egyptology, including learning basic hieroglyphics, and read books on the subject, but I have never come across reference to surviving Ancient Egyptian annals. As far as I know their history is reconstructed piece-meal from sources such as:

-Tomb and temple inscriptions (e.g. Ramses II's account of the battle of Kadesh in his temple at Abu Simbel; Harkhuf's late Old Kingdom tomb autobiography about his missions to lands to the south);

-Literature, some of which refers to historical events (e.g. the Tale of Sinuhe, which obliquely refers to the assassination of King Amenemhat I);

-The late 14th Century BC diplomatic correspondence found at Amarna

-Physical evidence e.g. DNA analysis of mummies and examination of their bodies for signs of illness or injury during life;

-and mention of Egypt in sources from outside the country e.g. the document known as the "Deeds" of Suppiluliuma I found at the Hittite capital Hattusas recording a request from an unidentified Egyptian queen (possibly Tutankhamun's widow Ankhesenamun) that the Hittite King send one of his sons to marry her, and what ensured.

The nearest thing to 'annals' that I know of is that there are references to a book, which does not itself survive but is quoted by other authors, from late in ancient Egyptian history, after the country had lost its independence, by an Egyptian priest who was known to the Greeks as 'Manetho'. His work apparently listed the Egyptian kings going right back to the beginning of their known history, with the length of their reigns, and grouped the kings into the numbered 'Dynasties'. Manetho was probably not totally accurate but he apparently had access to other sources now lost and his scheme of dynasties is still followed by Egyptologists e.g. many of the best known and most interesting rulers like Hatshepsut, Akhenaten and Tutankhamun all reigned in the Eighteenth Dynasty.