Like, olive oil?

I cannot speak to ancient Mesopotamia but oil was widely used in ancient Egyptian society. Regarding its culinary applications I quote from Egyptian Food and Drink by Hilary Wilson

The oldest royal food list, from the Sixth Dynasty Pyramid of Unas, includes five kinds of oil. Oil was an important food item as shown by its inclusion in the daily rations issued to royal servants like the Kings Messenger and Standard Bearer from the reign of Seti I. He received as his wages : '... good bread, ox flesh, wine, sweet oil, olive oil(?), fat honey, figs, fish and vegetables every day'.

One of the most popular oil-bearing plants in Egypt was the hegelig tree, also known as Egyptian balsam (Balanites aegyptiaca). Examples of the fig shaped fruit have been found in tombs of all periods including the Step-Pyramid. They commonly show a small hole bored into the flesh through which the oil-bearing seed was extracted.The sweetly scented oil was used for cooking and as a base for perfumes.

Another very popular oil, probably the 'sweet oil' from the Standard Bearers list, was that extracted from the seeds of the ben tree (Moringa peregrina). The tree is almost leafless and bears its edible nuts in pods. The Egyptians called the tree, and the oil obtained from its seeds, bak ,and the oil is commonly named in the offering lists of all periods. The tree is now very scarce in Egypt.

Oils which are now out of favor for culinary use were extracted from a variety of seeds including lettuce and radish, castor oil and linseed. the latter two oils may have been predominately for lightning while linseed oil is traditional used in Egyptian cookery, particularly for flavoring ful medames . Castor oil, though highly distasteful to western palates, is still used around the middle east as a cooking oil.

Floral tributes in burials contain safflowers, so oil may have been extracted from their seed, but most probably the commonest cooking oil was that obtained from sesame seed. There is little direct evidence for the ancient production of sesame oil but the plant was certainly grown in Egypt from at least the Eighteenth Dynasty, and quantities of the oil were imported from Palestine.

There is a possible reference in the Old Kingdom Pyramid Texts to sacred olive trees grown at el-Matariyah and in the Eighteenth Dynasty Ankenaten was shown presenting an olive branch with leaves and fruit to Aten. Olive trees were grown in Egypt, as shown by the leaves in funerary wreathes from Theban tombs an settlement sites like Amarna. Possibly small amounts of olive oil were produced but present a minor contribution to Egypts total oil requirement. Even as late as the reign of Ptolemy II, when comprehensive laws governing oil production were published, olive oil was not mentioned. Either the Egyptian crop was not sufficient or the fruit not suitable for pressing.

Nearly thirty types of oil are known by name from Egyptian sources. Some may represent the products of particular regions, including imported oils, and others may be blends or oils flavoured with aromatics or herbs. Not all oils would have been used for cooking and many have yet to be identified. even the remains in labeled jars are difficult to analyze.