That’s a pretty unfortunate situation you find yourself in. In case you still have access to the email address you signed up with, but don’t remember which one you used at the time, Facebook’s help page suggests the following behavior:
Ask one of his Facebook friends to look at the About section of his profile and send you the email address listed in the Contact Information section.
Try entering every email address or mobile phone number you’ve ever had (one at a time). You may have added an email address or mobile phone number to your account and forgotten it.
Enter your username. Your username is your personalized Facebook URL.
If you don’t know your username, ask a friend to go to your profile and send you the username in the URL.
Try entering variations of your name, in case you signed up with a nickname.
If you are prompted to enter a friend’s name, try a variety of friends if the first friend you try does not find an account. Remember to enter your friend’s full name, not just their first name.
If you do not have access to your email address either, perhaps because you signed up for Facebook using a 10 Minute Mail email address, or something similar, you have a bigger problem, but it is not impossible, if you have configured some of your friends to be ‘Trusted Contacts’.