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“iTunes won't let me access my backup because iTunes keeps asking for a password. I never remember setting up such a password to encrypt the backup. There is no need to protect it with password at all, so I don't know what password they want. And when I put my apple ID password in, it says it's the incorrect one. I am at a loss for what they want.”

Why iTunes Asking for the Password to Restore iPhone Backup?

After a search on Apple Support Community, I found that so many people suffered this random problem. Why iTunes will ask for this password? There are two possibilities. One is you do set up one when you sync your iPhone and create a backup to iTunes. The other is the bug of iTunes so that it will ask for the password randomly at backup restore. 

How to Get the Password for iPhone Backup Restore?

It is always so annoying when password is incorrect at backup restore. If you do set up an encryption for your backup, try to recall them carefully. You frequently used passwords are all the possible ones. There is a list below to give you some inspiration at your recall.

Trial One: Try Frequently Used Old Passcode

Apple ID password: Some people said this worked but others said it did not. 
iTunes Store password: This maybe the most possible one. 
Windows administrator password: If you are using a Windows PC and have set up the password for your PC, try this one.  
The "Passcode" you used to lock your iPhone: Or the passcode/lock code used on your iPhone when you first ever used the device.
All sorts of variations of your children's names and birthdays: Someone is likely to use baby’s birthday or names as the passwords.  
The ones suggested here: 0000, 1234, etc

Trial Two: Ingenious Tricks to Clear the Password

When you can’t meet the password with the passcodes you can recall, try the following settings. They may be help to clear the password on iTunes backup. 

1. Deleting the apple.com Cookies in Safari

The apple.com cookie contains the iTunes password and the Safari settings are linked to iTunes Store browser. So if you reset Safari or delete all cookies, this could be fixed under some circumstances. Open up the preferences and go to the privacy tab then change the "block cookies" settings. After that, you should be good to go. 

2. Reconnect Your iPhone with Computer and Login to iTunes Again

  > Disconnect your iPhone with computer by unplugging your iPhone digital cable out of the USB port.
> Sign out of your iTunes account.
> Connect your iPhone back to your computer again.
> Sign in again with your iTunes account.

3. Take a Cup of Coffee Then Try It Again

Why do I suggest this? Someone told me that iTunes would tell you that the password is wrong (several times over) when it is in fact correct.  Sometimes, it is the sever bug causing this incorrect show. So wait a moment and try it when sever is free and normal.

4.Try Something Like 3rd Party Application

Though sometimes we are lucky and the password problem got rid of easily with these suggested methods, we can be annoyed with obstinate incorrect password. So what is the most reliable method? Free solutions are always limited by finite 
power. So as the last choice, we can consider using paid tool to unlock the password. Have you ever heard of such a ware to recover lost or forgotten iPhone backup password? I remembered there is one named Tenorshare iPhone Backup Unlocker claimed to retrieve password for your iTunes backup. You can have a try, no personal experience, however. It is available on CNET and official site Tenorshare iPhone Backup Unlocker

iPhone backup password cracker
Tenorshare iPhone Backup Unlocker 

Extra Tip: How to Prevent iTunes Setting up the Password for You?

If you have set up a profile on your iPhone, you will need to go into Settings >General >Profile. Remove the profile, otherwise iTunes forces encrypted backups.

When you go to iTunes and make the backup, click on the iPhone in iTunes and you will see the backup settings. There is a box that says "encrypt iPhone backup". Tick it out if you don’t want your backup to be encrypted.  

That is all what I can suggest to get rid of your iPhone backup password. Hope it helps somehow.