First, how deeply assured are you that web programming is going to stay there for a long while? or alternatively, how long you
think you'll be able or willing to keep doing the job? Personally, I classify myself as a kind of programmer who likes to explore things. I like to play not only with the web stuff but writing drivers in Assembly, C++ Libraries, graphical programming, etc etc. As a consultant, I get to do various types of projects. Some require C++ knowledge; whereas others are fine with Perl. Overall, however, ability to think analytically and be a good problem solver is paramount in any work I do.
For one, I very much liked (and still do) Physics, Calculus, and Math. Naturally I did good in all of these areas (hack, my Physics teacher even suggested I pursue a career of a 'rocket scientist'; whilst my Calculus teacher insisted I pursue that of a statistician... ;). This, I believe, played out quite well for me in my work as a developer. Math/Calculus trains your brain to think in abstract fashion. Having a good grasp of precise sciences allows one to device better algorithms and data structures to employ in his/her code.
# Under Construction