Only three things I can think of:
1. Try booking through a travel agent. This might only work if you actually wanted them to arrange a real itinerary for you, not just a plane ticket. I've never used a travel agent myself, but I know that they can get better rates on things like hotels and tickets because they buy in bulk. You could try calling one in the U.S., but you might have better luck looking for a travel agent in Taipei, just make sure they're a legitimate company.
2. Fly standby. (In other words, don't buy your ticket until you arrive at the airport, and tell them you want to wait until some one for a flight to Taipei is a no-show, or they have a previously unforseen extra open seat for some other reason on a flight there.) I'm not sure this would work for internatioal flights, I've only heard of people doing it for domestic ones. Try calling the airport from which you're thinking of departing and ask if they do this. If they do, though, it'll probably still be brutally, brutally inconvenient for you, since I doubt and airport would have more than, at MOST, two flights to Taipei a day, which means you could be waiting, like, days at the airport for a flight with an open seat. (This is why I figure an aiport probably wouldn't offer standby seating for internatioal flights.)
3. Get a job in Taipei for the time you're there, and see if they'll pay part of your ticket. This would probably mean teaching English, but maybe not. (E.g., in the unlikely event that you're a girl and attractive, maybe you could get a job dancing a couple times during the week at a nightclub or something. I know some people do this for Japan, though I doubt their employers help with their ticket costs.) Not sure what the nature of your stay will be there, but just thought I'd mention this option for the sake of thoroughness. lol. It's probably just wishful thinking, from the sound of your post this isn't your goal.
4. If you're in college and a good student, try selling this to your college's office of international studies (or equivalent) as a short study abroad or research project, and see if you're eligible for public funding through your university. It would pobably help if you got someone from a university in Taipei to write a letter vouching for you. Lots of people do this, just depends on how much money your school has.
