Is there any way to recover your apple ID if you forget both your password and your recovery key?
This happened to me! I created a new Apple ID using my device (because it was the only way to create an Apple ID without having to have a non-iCloud email account associated with it), didn't realise that Find My iPhone was switched on, and then promptly forgot both the password and the security questions/answers for the account. I could still use the device, but I was unable to switch off Find My iPhone or erase the device. So I force-restored the device, but was then presented with the activation lock.

Apple helped me reset the activation lock because I was able to prove that I purchased the device. Your receipt contains identifying information such as the Serial Number and IMEI number, so when I provided Apple with this receipt, they were able to remove the activation lock for me.

But I never recovered the Apple ID itself.