A priori, it seems to me that one should engage in activities only if they satisfy at least one of these two conditions:

They are interesting: People receive direct hedonic value in the process of doing them.
They are important: People acquire relevant knowledge and skills that help them with other activities (some of that help could be through signaling).

But investigation we did in connection with our research at Cognito Mentoring led my collaborator Jonah and me to notice that a fair number of people seem attracted to learning things for the sake of learning, although they neither have an internal belief that learning the subject is important, nor a deep interest in learning that specific topic. Some of them then get frustrated that they're unable to make progress on their self-set learning goals, and this may harm their self-esteem (and put them off learning more important things later). Others may experience success that encourages them to learn more things, some of which may be interesting or important. Our page on managing your time generally advises against participating in and focusing on such activities, or at minimum critically considering whether the activities are sufficiently important to justify engaging in them even if one doesn't find them interesting.

However, Jonah and I may be missing important perspectives. I've heard claims that engaging in activities that are neither interesting nor important has intrinsic value -- it helps build character, makes one grow as a person, or it just might turn out to be important.

This school of thinking is reflected in diverse quarters. Tiger Mom Amy Chua famously forced her daughters to learn musical instruments to build their character, even though at least one of her daughters found it a terrible experience, and there was no reason to believe that the activity itself is important. The belief that one should try and learn new things is also widespread (albeit in a very different sort of way) in the rationalistic self-help community.

What's going on? Some possible explanations:

A mentality of trying and learning new things, including things that seem neither interesting nor important prima facie, might be crucial for learning new life habits and productivity hacks (see here and here for analysis by Dan Keys of CFAR that provides evidence in favor of this view).
Some people are very high in the openness to experience dimension. The very fact of trying out new things generates value for them, even if the activities do not interest them much at the object level.
Some of the "learning new things" is done in a social context, and the value is derived from the social context rather than the activity itself. However, the learning activity provides better structure to the social context.
Some people who don't have specific long-term goals don't have a huge number of important things to do -- at any rate, not enough to fill up all their time.
Some people are motivated more by a sense of short-term accomplishment at things that look hard than by the hedonic value of doing things in real time, so they gravitate towards activities that can best indicate short-term accomplishment to themselves or others.

What do you think?