"Why bother? Everything worthwhile has already been discovered." I don't think you understand just how wrong you are.
In astronomy, scientists are discovering planets that more and more resemble Earth. In our lifetimes, perhaps, one will be discovered that has evidence of oxygen (which strongly suggests life). Moreover, humanity is still working on ways to explore the inner solar system, and the person who rejuvenates space exploration will die rich and famous.
In biology, genetic engineering is pushing at the limits of what was thought possible. The most difficult diseases might be curable with some applications of genetic engineering: malaria, hemophilia, HIV/AIDS, etc. Moving to biochemistry, there might be ways to solve the energy crisis or slow global warming that are on the verge of being discovered.
In chemistry, there are plenty of molecules we know next to nothing about. One of them might be the secret to curing a painful disease.
In computer science, remarkable progress has been made in past years, but the dominance of the Internet will lead to exciting new paradigms and ideas.
In the environmental sciences, there is the problem of global warming (and also pollution, ozone depletion, peak oil, etc.; all of these problems are related). Do we have solutions yet? We sure as hell need to do something. And yet these are not impossible problems.
Moving to geology: we can't predict earthquakes and volcanoes yet. By monitoring seismic activity, it might be possible. Want to save lots of lives and be famous?
In mathematics there are many unsolved problems, and some of them are quite important to cryptography (i.e. the entire Internet relies on recent discoveries in math). Even seemingly simple questions, such as whether there are an infinite number of twin primes, are still unsolved.
In paleontology we are constantly discovering new fossils that reshape our views of the history of life. With new techniques it can be even easier to visualize the past... and perhaps even clone it.
In physics there are still many, many unanswered questions as to the fate of the universe and the nature of matter. Figure out the Theory of Everything and you'll be more famous than Einstein.
Then there is the question about how to fix the United States of America. It is unquestionably broken - but how should it become effective once again? And doubly so for Europe or Africa or India or China.
In art, there is no distinctive contemporary style and no incredibly famous living artist. You could fix that.
Music suffers from fragmentation, with seemingly more styles than there are artists. In classical, jazz, and rock alike, there is no unifying theme or composer, and lots of discord. Music will enter another great era, but who will be there to lead it?
In writing, there is lots to do. Unhappy that the entire young adult section of the bookstore is paranormal romance? Write something amazing and fix it. Want to be the first American in a long time to win a Nobel in literature? Your future is out there. Not to mention the importance of the Internet on literature, which is not completely known.
Now linguistics. It is predicted that thousands of languages are going to die in your lifetime. Can you save or record one, thus preserving an ancient and unique culture?
In political science, how can the Internet be used to promote - or suppress - democracy? Would true, informed democracy be possible if every citizen had a wireless connection?
Now if you like exploring... there are still plenty of summits not yet reached, and the vast majority of the ocean has not been mapped. The deepest point of the ocean might not even be known to man yet. You could fix that. And there are still many, many unexplored caves, too, just in case that isn't enough.
In archaeology, there are still plenty of new discoveries coming in. Do some fieldwork in the right place and you'll chance across a new Amazonian city. Theories of the past are being rapidly changed.
There is so much opportunity out there in the world. Sometimes, I think it could not be more exciting than to live today.
Here's Steve Jobs to back me up (46 seconds):
Information gleaned from tick_tock_clock at Reddit.