This Response Post is Going to be Legen- wait for it...

     The rule of three is very common across the different areas of study.  It is especially common in english, where groups of three are seen consistently. There are the three bears, little pigs, and billy goats gruff. Ebenezer Scrooge is visited by three spirits. Its seen as the minimum needed to establish a pattern and is short enough to be memorable.  In math you're taught to graph using three points to establish a trend. It even appears in law and the arts in a similar manner.
     What makes it different in comedy is that the first two typically establish some kind of pattern for the third one to break in some unexpected way as to produce laughter. Its like an extension of the incongruity theory.  Where the first and second examples are setting an expectation for the last one to let down. The rule of three might not seem entirely necessary in many examples as the third is incongruous on its own. Barney simply saying "a hug is like a public dry hump" is certainly not something the average person would say making it incongruous in that sense. However, Robin's two examples before that set up a benign aside about how to end an encounter with an ex, making Barney's statement stick out even more.
     Barney himself is a very strong example of another trend in comedies, main characters that are pretty bad people. His character is defined by having sex with women under false pretenses, behaving like a spoiled child, and putting any chance with women over his own friends. Thats what makes him funny to the audience. The rest of the people on that show have unsettling flaws of their own.  Maybe its an incongruity thing with them acting contrary to what it socially acceptable. On the other hand it also links to the superiority theory.  We think we know better than these people so we laugh at how shallow or broken they are. The relief theory could have a hand in it just as well.  We laugh because we see people acting in ways that we ourselves might if it weren't so discouraged by society. Its a cathartic thing.  Either way, you see it in almost every show from How I Met Your Mother to Friends to Always Sunny in Philadelphia, going back to Seinfeld and even further back to shows like the Honeymooners. It seems as old as the comedy itself.
 Speaking of catharsis...

-Dary