The kitchen towel became completely oil-soaked from wiping the greasy pan repeatedly.

oil-soaked 是由名词 oil（油）和动词 soak（浸泡）的过去分词 soaked 组成的复合形容词。oil 表示"油"，soaked 表示"被浸泡的"，合起来表示"被油浸透的"。