There are a number of annual academic competitions that offer scholarship money as prizes. Below is a list of the most prominent independent competitions, their deadlines, and where you can find out more about them. Keep in mind that these are all done on a yearly basis, so if you missed the deadline this year, there's always next year!
UNA-USA Essay Contest - Deadline: January 5th - One of the most prestigious national essay contests available. For this contest, you are asked to write a letter to the president about a topic related to U.S. international relations and the United Nations. The topic changes every year but it's usually related to development and the place of the U.S. in the UN. Top prize is $1,500 and a paid for trip to be a guest of honor at a meeting of the U.S. chapter of the U.N. in New York. There are also regional winners and semifinalists, so if you put a real effort into this contest, you can very realistically get some recognition. I would recommend as someone who was a national finalist, that you first do a lot of reading from respected publications such The Economist, the Washington Post, WSJ, etc. because the judges really like an extensive and diverse bibliography. If you like objective persuasive writing with a research aspect, this can be a very fun contest because I know it was for me.
Ayn Rand Essay Contests - These essay contests ask students to write interpretive essays on one of three topics for three of Ayn Rand's books: Anthem (Deadline: March 20th), The Fountainhead (Deadline: April 6th), Atlas Shrugged (Deadline: March 20th). The Fountainhead and Atlas Shrugged contests each have 236 total winners and offer $43,250 in award money and the Anthem essay contest offers $14,000 in award money.
JFK Profiles in Courage Essay Contest - Submissions accepted September 1st - January 9th - High school students are asked to submit original essays about a political figure that demonstrated political courage. Total awards total $13,500 and the winner is flown to accept the award at the JFK Presidential Library in Boston.
Society of Professional Journalists Essay Contest - Deadline, March 5th - This essay contest is interesting because the topic stays pretty much the same year after year and because the word limitation is rather small. Therefore you really need to utilize concise, powerful writing to do well. One of the best things about this scholarship is that it has state winners that go on to national judging to compete for a couple thousand dollars of award money. However, due to the relatively unknown status of this competition, winning at the state level is very doable. When I entered this competition, I spent a total of a week and half doing research and writing and ended up winning the state of Illinois. While I didn't win at the national level, the director of the Illinois program at the University of Illinois said I could come visit him to talk more about it and I'm sure that had I been on the cusp of getting accepted there, something like this would have definitely gotten a nod from him to the office of admissions...
Dupont Challenge Science Essay Contest - Deadline: January 31st - This unique essay contests let you write an essay expressing your views on a variety of scientific topics ranging from geophysics to astronomy to environmental science. Your essay should be well researched, cite scientific journals, and make a compelling point about the future of your chosen topic. Split into two age groups, and offering a $5,000 first place prize, this is a great opportunity for those of you interested in science.
College Scholarships Blogging Scholarship - This one is pretty straightforward. Write and maintain a unique, interesting, and well followed blog and you can win the $10,000 first place prize or several runner up prizes. Not only is this serious money for doing something most people do for next to no money, you can monetize your blog with ads if you build a good following.
American Legion Oratorical Contest - Deadline: March 8th - This competition asks students to prepare and deliver a speech about the Constitution, citizens' duties, and our obligations to the government. Speeches must be between eight and ten minutes and cover one of the assigned topics. To qualify for the national circuit, you must win a state level departmental contest, at which the American Legion will pay for you and your chaperone's travel expenses. Over $138,000 of scholarship money is awarded each year so if public speaking is something you excel in, give this a look.
National Peace Essay Contest - Deadline: February 1st - Win up to $10,000 as the national winner or $1,000 and a trip to Washington as a state level winner. The essay is a three part contest on the topic of peaceful civic action and international peacemaking. Open to all U.S. high school students; see website for complete contest details.
Prudential Spirit of Community Awards - Deadline: November 1st - A competition designed to recognize outstanding volunteer community service whether it be through leading and organization independent projects or donating one's time. National honorees can win up to $6,000 and $5,000 for a charity of their choice. While it may not ask you to do anything new, it can be a great way to hit two birds with one stone and receive money for volunteer work that you should be doing anyways.
Toshiba ExploraVision Awards - Deadline: February 2nd - A science competition that challenges students to use their imagination and scientific knowledge to create a new vision of future technology. Each team needs an official coach and must complete an application that includes scientific sources. Four teams can win $10,000 or $5,000 and a number of unique Toshiba products and incentives.
Young Naturalist Awards - Deadline: March 1st - A scientific competition that asks students to inquire into a scientific topic, do independent research, and come to a new understanding of that topic. You can win up to $2,500 as a first place 12th grade winner.
You'll notice that most are independent writing competitions. This is because the worthwhile math, science, and music competitions are typically run through school structured clubs and nationally organized competitions. Seek out scholarship opportunities within your school network and extracurricular connections to see what competitions exist through the school for more opportunities.