You can access this super useful page at http://supportprofile.apple.com, and all you have to do is log in using your Apple ID, and any hardware you've registered using that Apple ID will appear as if by magic. You can manage this page by adding or removing hardware, and that's basically anything Apple has sold in recent memory; my list dates back to my iPod Shuffle from 2005, but Apple can keep track of even older harware than that:
Behold the oldest things in Bryan's Support Profile!
You can also view each item and see more information about it, such as the warranty status and support cases. And if nothing else it offers a handy reference to the support resources for that hardware, meaning knowledge base articles for specific issues and even just general troubleshooting:
Support Profile details for Apple TV
This doesn't necessarily have to be the same Apple ID used on the device to sign into the store or any iCloud services, so an easy way to keep the list up to date would be for one Apple ID to be the “main” one used to register new hardware so that you have all of it in one central location, including repair history and serial numbers and purchase dates. This makes it that much easier when you need the information for any reason.
It's also a good idea to keep this list current, so if you aren't even sure where that old iPod is anymore, or you still see the computer you had three computers ago, you can also remove devices. Helps make the list a bit shorter and also you don't have to slog through a bunch of extraneous data to get to what you want.
At least for me the best part is that I didn't have to “do” anything to have all this data, it's just there because I registered using the same Apple ID for all those devices. And if I ever need a purchase date or have to check on the status of support for something, doing so is a piece of cake.