The prisoner's dilemma is very useful as a beginner/layman introduction to the game theory mindset. It has a similar role arithmetic has in preparing you to calculus/differential eqs.
Volunteer's Dilemma and the Nash Bargaining Game have more complex equilbiria, so they're not quite as good if you're looking for an introduction to game theory. Stag Hunt is basically a game about whether the players know they are in such a game and can communicate about it, so it's not the best for learning about games themselves. Prisoner's Dilemma really stands out among the basic games.
But really, Prisoner's Dilemma is uniquely fascinating in iterated form [0]. Iterated Stag Hunt would be boring, and Iterated Volunteer's Dilemma would be crazy, but Iterated Prisoner's Dilemma is a very deep game which seems to have things to say about Biology, Politics, and Computer Science all together.
Sure, it's a standart problem in game theory, but it's "uninteresting" to say the least
For example, it has a limited Nash Equilibrium http://en.wikipedia.org/wiki/Nash_Equilibrium
It also is rarely applicable to "real situations" like Economy, populations, etc.
Sure, the wikipedia page has various examples, but in real situations it's usually something else, as there are more details