How to Buy Bitcoins Anonymously

For those just getting into Bitcoin, the first step is the hardest - figuring out how to buy bitcoins.  The Beginner’s Guide to Bitcoin explains that you have to make a choice when buying bitcoins. You can buy them quickly, you can buy them cheaply, or you can buy them anonymously - but you only get to choose one of those options.
This is the second in a three-part series about buying bitcoins. The first part can be read here - How to Buy Bitcoins Quickly.  The third part, How to Buy Bitcoins Cheaply, is coming soon.
One of the great advantages of Bitcoin is the ability to stay anonymous when making purchases.  Because no information is stored anywhere identifying who owns which Bitcoin address, there’s no way to know who controls which bitcoins.
But that anonymity can end quickly if you’re using services like Mt. Gox, Coinbase, or any other automated method to purchase bitcoins online.  They all require you to provide identifying information, which you have to assume is stored forever.  Should a government want to get that information, it’s only a subpoena (or your country’s equivalent) away.
If you do purchase your bitcoins through one of these services, you can “clean” the bitcoins by sending them through a service like BlockChain’s Send Shared, but there will always be a record of you having made the purchase.
If you want to retain a higher level of anonymity, you’ll have to avoid any of the automated purchasing routes, and stick to something more manual.  If you buy your bitcoins directly, either online or in person, you can get them without ever revealing who you are.
Local Bitcoins
It may take you a bit out of your comfort zone, and it is a bit strange to be buying an electronic currency from someone in the real world, but buying bitcoins a seller nearby is probably the fastest way to get bitcoins, and it also lets you stay anonymous.
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You can use LocalBitcoins.com to find a seller near you.  Sellers on the site typically list their bitcoins for sale at some price above the Mt. Gox exchange rate, and can also specify upper and lower limits of money that they’re willing to change into bitcoins.
Always check out the profile of each seller to see how many previous sales they've made and the feedback score from previous buyers.  You don’t want to be bringing a lot of cash to meet with someone with a bad reputation.