Update: Changed wording that explains whether the developer or Google Play will be responsible for your refund. (Thanks stlc_8tr.)
Update (09/11/2014): Google has now expanded the refund window to two hours. If you'd like to request a refund after the two hour window, the steps below are still valid for starting the refund process. Check out Scott Webster's post for more information on Google's policy change.
Have you ever bought an app that you meant to try right away, but then got distracted by something else? Now that your 15-minute grace period has run out, the cost of the app is your responsibility. While this is not a big deal with apps in the $2 range, you may not feel the same when it's $10 or more.
In the past, you would need to email the developer and ask him/her/the company to issue a refund. Unfortunately, some developers just ignored those requests, causing some misdirected anger at Google. As a result (or so it seems), Google is now handling refunds on their own after that 15-minute window. Ready to request a refund? Let's get started:
Step 1: Head to Google Play in your Web browser. Click on the settings cog in the top right corner and choose My Orders.
Step 2: Find the app for which you want a refund in the list and move your mouse pointer over it. You will see a button with three dots appear; click the button and then choose "Report a problem."
Step 3: In the pop-up window, choose "I'd like to request a refund" from the drop-down menu.
Step 4: Fill out the reason you think a refund should be issued and click Submit.
Step 5: Check your email for a notice that Google has processed your refund request. If it's not there right away, give it at least 48 hours before you try to take any further action.
According to the experiment AndroidPolice conducted, the developer does not receive any communication about the refund request or process if it takes place 48 hours or more after purchase. To think that Google is footing the bill for the developer seems to be a reasonable conclusion.
Google probably has some built-in protection against people trying to abuse this system -- so only request a refund if you really have to.