Developer Aaron Straup Cope has released Parallel-ogram, a web app which archives your Instagram photos and likes to make it far easier to look back at them later. The app uses Instagram's API to monitor all of your activity on the site, both private and public, and creates your own personal photo stream. For now, all of your content stored by the service is private — there's no built-in sharing — but it provides a service that Instagram has been sorely lacking.
The app isn't hosted — you'll need to install it on your own computer or a web server, though instructions are provided which should make the process a whole lot simpler. Straup Cope has history with this sort of software — he's created similar apps before for Flickr and Foursquare, allowing users to take more control over their content, and acting as a constant takeout service. If you'd like to give it a try, the project is available from GitHub now.