So far, all of the following have been suggested as really important, either more important than the others or at least as important. In alphabetical order:
And number theory got an honorable mention for being awesome.
How are people judging what is more important than what? I hope not by looking at how often they use these things themselves.
Not a month goes without my having occasion to use all of these, and 3D geometry, and other mathematical stuff as well. But as an applied mathematician and programmer, I would, and they all seem to me like just a step beyond basic arithmetic. It would be easy, too easy, for me to say that everyone should know this stuff, so I won't.
Instead, FYI and FWIW, here's the current high school curriculum in the UK for A-level maths.
This guy says that the problem is that high-school math education is structured to prepare people to learn calculus in their freshman year of college. But only a small minority of students ever takes calculus, and an even smaller minority ever uses it. And not many people ever make much use of pre-calc subjects like algebra, trig, or analytic geometry.
Instead, high-school math should be structured to prepare people to learn statistics. Probability and basic statistics, he argues, are not only more generally useful than calculus, they are also more fun.
I have to agree with him. What do the people here think?