请客 is a verb-object-verbal phrase (联合词), therefore it is able to seperate both parts and put further structures in between. I read I could put 起 in between together with 来 at the end to form a so called "complement".
Now, all i know that i can build such a structure, for instance:
-分类 --> 分起类来
-吃惊 --> 吃起惊来
-贬值 --> 贬起值来
-游泳 --> 游起泳来
the problem is, i only know how to build this structure, but i can not find any explanation how the meaning is changed by it.