The two systems of cognition a la Kahneman each seem to each have roles that they fulfill better than the other and in which, for optimal performance, the other system ought not to interfere. Until recently, I had only really thought about when System 2 ought to override System 1, but I believe other cases are worth considering as well.
It is important to be able to get System 2 to shut up and let System 1 work in certain circumstances. Physical movement is an excellent example of this. When you shoot a basketball, throw a football, or use a video game controller you may notice that when you are “on fire” you are not focusing at all on the mechanics of what you are doing. In fact, if you start thinking about what you are doing better than usual and begin analyzing the mechanics of the movements, you will likely begin performing worse. Similarly, when you focus on some specific motion that is supposed to contribute to a correct movement, applying the conscious effort of System 2 to force yourself to do it, you will generally exhibit worse performance than if you just hold an image in your mind of what the movement should generally look like.
The override of System 2 causes you to neglect other important factors that go into a movement: and there are too many for system 2 to think about at once. The strength of System 1 is that it can handle all those factors without trouble.
Thus, we can see where it is important to learn the skill of not trying too hard or focusing conscious attention too forcefully. We often cannot consciously keep in mind all the necessary factors, though we are capable subconsciously, as long as System 2 does not interfere with the process by introducing an override in one specific area. While learning this skill, it is important that one not try too hard to not try too hard. Thus, it can be a very difficult skill to learn, I think particularly for analytical people. See The Inner Game of Tennis for a more thorough exposition of this idea.
The skill of effortless effort, is a very useful one, though, I think, not quite as important as learning how to get your mind to actually try, with System 2 reevaluating the lazy heuristics of System 1, which is one of the necessary skills for long term rational decision making. System 2 overriding System 1 is necessary for dealing with many cognitive biases. To avoid allowing representativeness to alter my judgment of what someone does for a living based on a description of them, I must avoid the impulse to merely match the description with a relevant stereotype and output an answer. System 2 must recognize the need to apply base rates and Bayes Theorem, and remember that base rates are probably better evidence of what someone does than the stereotypes that mesh best with a possibly unreliable description. LessWrong is full of other examples of when this skill is important and why.
Then there is the skill of using both systems in unison, each doing the job it can do well and not interfering with the job of the other system and worsening overall performance.
Take a martial artist as he fights his opponent. He must allow his System 1 to handle the mechanics of how his body moves—the complexities that he has trained for throwing a perfect punch are far more numerous than he consciously is aware. If he focuses too hard on one particular aspect of the strike, such as proper shoulder rotation in his punch, he may lose fluidity, speed, or power. System 2 must not focus on how he performs each strike. At the same time System 2 is very active, evaluating his opponent’s movements, her favored strikes, how she responds to feints. System 2 creates plans for exploiting any perceived weakness. And in this area System 1 must not enter, System 1 sees a fight and becomes angry, abandons caution and begins using too much power, a mode of fighting that will be quickly punished by an experienced opponent.
When the two are in balance, the martial artist will experience a state of transcendence, other concerns falling away as all attention is focused on the challenge in front of him. I have found this sort of experience tends to provide the greatest fun density per time as compared to anything else I do.
I am not sure if this is the most important of the skills, I think getting System 2 to rethink System 1’s overly hasty judgments is still likely more important if one has goals other and higher than personal enjoyment. But getting both systems working together in unison feels the best of any of the skills and I think engaging in activities that allow for both to be used may be an excellent way of increasing the amount of fun you have in the limited time you have to spend.