Depending on where you purchased the game, you can retrieve your game in the following ways:
Steam:
If you purchased Don't Starve through Steam, the game would have been automatically added to your Steam game library (as it would normally). Log in to the Steam account that you used to purchase the game, and you will find the game in your Steam library.
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Humble Store:
When you purchase Don't Starve through the Humble Store, you will be sent an email that contains a unique URL allowing you to claim your game installer.
Where is my email?
If you either A) mistype the email address or B) type in a different email address than you think you used to purchase, you might miss the purchase email confirmation.
If that happens, your best, next step is to check any possible email address (and spam box) to see if you can find the Humble Store email confirmation.
You can also go to the Humble Bundle re-sender tool to request for the purchase confirmation email to be re-sent to the email address you used to purchase the game: https://www.humblebundle.com/?s=resender
If for some reason you are still having trouble retrieving your game installer or account, contact customer support with your account information and/or your transaction information and we can help you out.
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GOG.com:
If you had purchased Don't Starve from GOG, log into your GOG account and check the Account Orders & Settings page to find your order history and the game. https://support.gog.com/hc/en-us/articles/212184469
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Chrome Web Store: (During the Don't Starve Early Beta)
If you purchased Don't Starve through the Chrome app during the early beta and were not able to translate your Chrome key into a Steam key, contact customer support with your order number and the email address you used to purchase the game.