The title abu Amurrim means literally “father of Amurrum”. It is attested for five rulers of the Old Babylonian period: Kudur-Mabuk, Warad-Sin, Rim-Sin, Hammurabi and Ammiditana.
There are also two published texts in which it is not certain whether the bearer of the title was a king or, like Kudur-Mabuk, a ruler of equivalent status. UET 5 62 is quoted in full below; for the moment it will be sufficient to note Kupper's plausible suggestion that in that text the person referred to as king is Warad-Sin, whereas the abu Amurrim is Kudur-Mabuk.
Kupper has also suggested that in PBS 8/1 79 the abu Amurrim is Ipiq-Ištar, the king of Malgûm. The text records a MU.TÚM delivery of two head of cattle by a certain Ipiq-Ištar, AD.DA MAR.TU. Here there is nothing in the context to suggest the latter had royal status; the only reason for making the suggestion was that elsewhere the bearers of that title were either certainly of royal status or, as in UET 5 62, probably of royal status.