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From An Introduction to the Theory of Surreal Numbers:

[...] Thus the reader has the opportunity which is all too rare nowadays of getting to the surface and tackling interesting original problems without having to learn a huge amount of material in advance.

What other math fields would be in the same class as this one? What other math fields wouldn't require learning a huge amount of material in advance and would allow one to tackle interesting original problems?

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Well, pretty much by definition: Any subject taught in first or second year university. –  EuYu Feb 20 '13 at 6:59
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Yes, my bad. I edited the question. –  Vladimir Putin Feb 20 '13 at 7:03
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Some areas of combinatorics can be quite accessible (though its still no cakewalk). Perhaps someone with more knowledge can comment on this. –  Potato Feb 20 '13 at 7:07
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In my experience, most anything in linear algebra. For example, many data compression techniques require a relatively elementary knowledge of linear algebra, yet are very interesting, and have powerful results. Many such techniques still have open questions regarding the limits of simplifying their computational complexity. –  Andrew Jean Bédard Feb 20 '13 at 7:23

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Ramsey theory on the integers has research problems in every chapter (presumably still open) and I think it's pretty accessible to most levels.

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