What careers not requiring an academic degree might be open to a self-learner who wants to do research on pure mathematics?
Be very wary of doing mathematical research on your own. Even if you read all the books there are, you will most likely go off on a tangent and reinvent a theory that has been known in math for decades or centuries - simply because you never heard about it and did not know the commonly accepted names of the structures you have been working with. Or, as others point out, you might work on something that may be true but is of zero interest to anyone else.

If all you are looking for is the joy of doing mathematics, that may well be fine (although even then I'd say using up-to-date tools and theory would probably be more satisfying, just as I'd prefer to code on a modern machine and not an antique, even if it does run a compiler). However, if you want to make a difference in the mathematical world, possibly publish your findings, then you will need to interact with other mathematicians.

And this is where a formal education will help. A math degree will show other mathematicians that you at least have learned the fundamentals. When you contact a mathematician, you will need to differentiate yourself from a random crank. There are far too many home-educated random cranks in mathematics, and you are lucky if they self-identify by claiming that they can solve the general quintic equation by radicals. Having a degree in math will make it easier for you to get other mathematicians to talk to you. In addition, your advisor will have decades more experience than you and will be able to steer you in fruitful directions.