From my perspective, we could preliminarily investigate the following pair of questions:
- Are there any areas of mathematics that a self-taught undergraduate can successfully learn?
- Which part of mathematics produces the greatest number of open problems (since there are only a finite number of mathematicians out there, the greater the number of problems is the better your chances to produce something new in that open field)?
We can certainly answer affirmatively to the first question, but we need to agree on the semantic meaning of "successfully learn" and, IMHO, it is the level of knowledge that will allow the student to write a publishable paper on any non-predatory journal.
About the last question, by just taking a look at Open Problem Garden, we see that about half of the listed problems are in graph theory and combinatorics is also one of the "easiest" areas of mathematics that you can self-learn!
Generally speaking, I would say to try to "specialize" yourself in something that you like the most and that you can manage by yourself, at least for the next pair of years, and also to start climbing the ladder of mathematical knowledge by attending courses on a possibly different path, taking your time and starting writing papers at the same time on the "simpler" topic that you independently approached years before.
This would lead you to get some peer-reviewed publications when you are going to start your solid research on the main topic that you are mastering by following the ordinary path at university. Just my two cents.