There's always more to be said about archives in the ancient Near East, but I wrote about Hittite archives in What are our sources for the history of the Hittite Empire?, and u/Daeres discussed Neo-Assyrian archives in the AskHistorians podcast episode "Assyrian State Archives."

The genre(s) of texts attested in an archive depends a great deal on the time period and especially the specific site under discussion. The vast majority of texts from the Ur III period (ca. 2100-2000 BCE) are administrative and economic texts, for instance, whereas texts from the Neo-Assyrian period are more varied and include treaties, personal letters, literary texts, court records, omen texts, historical annals, and so on. Similarly, the distribution of genres varies enormously between sites. The city of Nippur in southern Iraq has yielded copies of nearly all of the known Sumerian literary texts, for example, whereas archaeologists digging at the contemporary city of Ebla in Syria have primarily found administrative and economic texts. It should be emphasized that the tablets discovered at each site are often not representative of the site as a whole but rather reflect the dates and functions of the buildings excavated thus far by archaeologists. (In other words, you're going to find different sorts of texts in an Old Babylonian house than in a Neo-Assyrian temple.) A mere 3% of Babylon has been excavated so far, and many Mesopotamian sites have received much less attention – if any at all – so there is still a lot to be found.

Marc Van de Mieroop has a good chart with a breakdown of the relative popularity of text genres by time period in Cuneiform Texts and the Writing of History. I uploaded a copy of the chart here.

For further reading, see Archives and Libraries in the Ancient Near East, 1500-300 B.C. by Olof Pedersen and Libraries Before Alexandria: Ancient Near Eastern Traditions edited by Kim Ryholt and Gojko Barjamovic.