What can be included in between a separable verb? I only know that a verb, which is made up of a verb and object, can be separated and further structures can be put between.
For instance: 发愁 , wherein 发 is the verb and 愁 is the object, can be separated and filled with an aspect particle and/or measure word.
-发了愁 (aspect particle)
-发个愁 (general measure word)
-发过四次愁 (aspect particle and measure word for occurences)
What else may be placed in between?
Here is what AllSetLearning has to say about separable verbs:


  ...until you understand which verbs are separable verbs and how they
  work, you'll forever be making mistakes with these verbs, even in very
  basic sentences.


BTW I think it is a quite decent summary.

Besides 了, 着, 过, and measure words, you can put other things in between if the verb is not "rigid". 

Some separable verbs are more rigid than others. For example: 发愁, 散步, 聊天, 分手, etc. Notice that the meaning only emerges when the two characters combined together. If you separate them as verb + object it just look weird.

Less rigid cases include 睡觉, 吃饭, 撒谎, etc. In these cases the verb and noun can be separately used with no problem, so you can say 我睡了一个美美的大觉, or 他撒了一个神仙来了都遮不住的谎.