It would definitely get frustrating having to peridiodically enter passwords for 10 email accounts. I found an article that has some suggestions if Mail is not remembering your passwords:
Mail (Mountain Lion): If your password is not saved
If you click “Save password” in a dialog in Mail but still have to enter your password, there are a couple of things you can try.
- Open Keychain Access, and then locate and delete the mail account password item.While in Keychain Access, you can also choose Keychain Access > Keychain First Aid to verify and repair keychain settings.
- Open Disk Utility, and then select the volume (for your hard disk or disks) in the list of disks and volumes. Click First Aid and then click Verify Disk. If necessary, click Repair Disk. Then send yourself a test message.
You can find the full article here:
Thank you for using Apple Support Communities.