Everything interests me, superficially[1]. Most of it is useless except for playing trivia games.
If I were a FU-rich autodidact that wouldn't be a problem but I'm a poor schmuck with a family to support.
My brain is so full of interesting stuff and tangential thought that there no room for focused attention on anything mundane enough to count as useful work (though I'm sure with an equipped workshop/lab and a couple of years I can turn one of my book-of-ideas concepts to something useful for mankind).
Don't we \have\ to do what gives us a living?
[1] superficially perhaps isn't the right word, the minutiae of everything is infinitely interesting too, more that things are only interesting temporarily, which leads to flitting about across diverse subjects.
So I sat with a pencil and a sheet of paper and wrote throughout the paper things that interested me. Then I linked all the related things together with lines, and ended up with my graph of interests. Things that, on contrast with those already in the paper, were too supeficial, I didn't write; and I wrote about 30-40 of them.
It was more illuminating that I initially thought! As I now had a clear mental image of how all interacted, and where the center of my interests laid. So little by little I started seeing the more fringe interests as what they were, and even as a function of their "path from the center". I started dedicating more time to the central thesis, and the fringes helped me gain new perspectives on the main "leit motiv".
And eventually I had an idea that resulted from all of that. Now it is the basis of my main side project. I do play from time to time with the rest, and the center is moving around a little bit, but there is a strong focus on reaching the goals that most interest me.
(Sorry if this ended up a little bit too abstract and vague; I hope it helps!)
Overall it's a 40 hour work week along with 3 extra hours of commute a day, so I take to reading mathematics/cs literature on the train and use the rest of my day to either relax or to learn more things.
I would say that while we do have to do what gives us a living, there are options. Contracting, for example.
I should say as well: I also have the same dilemma, I'm interested with everything. So I decided to choose for subjects (such as mathematics) that will give me a better base to learn more subjects.
Of course, nothing says you have to have only one job. Luckily, all of my interests are viable professions, so I try to do them all.
I have become especially fond of the latter approach, when possible. No million dollar moonshots like you often read about around here, but I've had good luck at least making back the cost of my time/materials, which is quite gratifying for something I'd tinker with for free in my spare time anyway.
I think it works really well for me. I love the flexibility and variety while still being practical. But, again, I have been fortunate to be almost exclusively interested in things that people are willing to pay for.
I had a roommate in college who every next morning after waking up talked of different interests/career options. This advice works for people who know what they are doing but, it's quite broad for people who have no clue what they are doing.
Each new piece of knowledge you learn will give you a better base from which to learn more, and slowly the amount you are able to learn will increase to help you cope with the load.
After a while you gain the ability to reduce a problem you're faced with down to other problems in other disciplines, then things start getting boring because you can already figure out a way to reduce this problem, etc. So at that point it's time to mix it up a little and refocus.
Another thing that should be noted is that you should always make sure that you are out of your depth with at least one thing you are studying. You can only really improve by pushing yourself. However remember that you cannot push yourself constantly, sometimes you need a break. So in doing this, you should be driven by your own interest.
What I have found is that I am not necessarily able to do everything at once, so I end up doing a rotation of things I find interesting at that moment. Eventually I'll either discard some topic or problem or such, because I don't find it interesting or I will find something new that I find more interesting. If things get stagnant, mix it up a little!
I've been doing this for approximately the last five years, and I think the payoffs have been great, and I have learned so much more than I think I would have otherwise. However I have nothing to compare to! So we cannot be sure =)
Do what interests you.