What is the best profitable small business a 13-year-old can open?

I congratulate you on your entrepreneurial spirit, few 13-year-olds would want to start a business. I'd like to wish you best of luck. There are a few things that 13-year-old businessmen are better at than fully grown ones.  One of those things is understanding exactly what other 13-year olds want (particularly if you have good interpersonal skills).  And that naturally means that one profitable small business you could set up is an online retail business.  And sell items targeted at people your age, and create a marketing campaign around that using words they want to hear. Online retail is a great starting point, a lot of people get into it because it costs almost nothing to set up, all you need is a website, a paypal account, and something to sell. Nowadays, China manufactures a huge number of things, that you can pick up at low low prices, and it's very very easy to purchase bulk quantities thanks to sites like http://www.aliexpress.com But of course, beware of less-than-honest sellers; low quality goods; and the various laws and regulations (including possibly stuff about tax). Comb through sites like that, and identify things you could sell, start with low-value items as this will minimise your risk.  Set up a website (even if you use eBay, you should set up a website), and payment systems (you may need an adult to do this due to laws and regulations about letting 13-year-olds access financial services).  Then use your interpersonal skills to get mentioned in the media to increase your exposure, making full use of the "awwwww, isn't he cute" factor that all adults uncontrollably express to any news about 13-year-olds wearing a suit, particularly ones with their own company. This is exactly the type of heart-warming, feel-good news, that the media absolutely love, so once you gain that exposure, it will have no problems spreading across the country. Suddenly you have that exposure, people will visit your website, now unfortunately at this point you might have to deal with most peoples' reluctance to purchase things from a 13-year-old, you might need to spin some positive PR about that by saying "purchases processed by adults, your payment is protected by Paypal buyer protection", that kind of stuff to put people at ease about the legitimacy of your site. You may be approached by people claiming to want to help you, be very careful about these, some may be fraudulent, but it would be bad to miss a great opportunity, so look out for them. Other than that, keep good financial records, make sure you understand the relevant laws and regulations (particularly taxation, and minimum age limits), always seek advice if you're unsure, don't sign anything.  And needless to say, you will probably need to seek your parent/guardian's permission on pretty much everything that you do. I think that's about it.  Be ambitious, don't for a minute accept that you can't achieve success just because of your age.  Learn fast.  Listen to advice but use your own judgement (and where you don't have enough experience to make the judgement, do some research).  Be weary of those who might want to exploit you.  Overall, have fun. I have a 14 year old brother, who at one point tried to start a retail business, which failed because they didn't have a sound business plan (and he ignored my advice).  I am in my mid-20s, I'm an engineer, and I have my own start-up business making micro UAVs and quadrotors, with headquarters in the USA and UK.

Buy a few packs of Koolaid and set up a small stand at a busy street/intersection. Fill up a cooler with ice and create large batches, then sell it for $1/glass. Huge profit margins, little overhead. Involves more of your time, but it'll teach you the basics of finance, negotiation, sales and marketing in a short while.

You can start a lawn mowing and home window washing business. Right now in your garage there is a good chance you have all supplies you need. If you live in an average neighborhood you can charge $25 for the front and $25 for the back depending on the size of each yard. Now mowing yards is not  anything new for a teenager, but most people do not wash the outsides of their windows on their homes.  You could charge $2 to $10 per window depending on the neighborhood and the difficulty cleaning (second floor, bushes, etc). So now you have two services you can offer for every house you approach with the potential to charge up to $150. Then when you get enough clients, you hire some of you friends to help you do the work. Now you have become the manager and making more money and doing less of the labor.  Good luck. 

At your age, I recommend investing in stocks and bonds. The time invested in understanding markets leads to better investments, and better investments lead to profit. It is also a business that works while you go to school.You can use coding to reach conclusions on when to buy or sell a little sooner, and you can use your interpersonal skills to sell derivatives to class mates, allowing you to expand your business. But first talk with your parents about legally registering your activities before you make any sales.Best of luck kid.