Of the many things your phone can do, automatically recording a phone call is not one of them. Perhaps not including that feature is a security precaution for phone makers; it could be a potential liability issue. Regardless, from time to time, documenting a call is a useful thing to do, whether you’re conducting a phone interview or cataloging a customer service call.
If your iPhone has support (via your mobile carrier) for 3-way calling and Visual voice mail, you have an option for the cheapest workaround of all.

I want to use a downloaded custom MP3 file as my custom greeting. Is there a way to transfer it into the file or do I have to just re-record it by hitting record while the MP3 file plays off of another device?
It only take a few minutes to set up your Voicemail account. Once setup is complete, your Voicemail will be ready to receive messages. Call 123 to connect to your voicemail. If prompted for a password the first time you access your account, use the last four digits of your mobile number. When prompted, create a password which can be any 4 to 7 digit code. (A good password is one that’s easy for you to remember, but hard for others to guess) When prompted, record your greeting and name. Your voicemail is now ready to use. Call *86 to connect to your voicemail. If prompted for a password the first time you access your new account, use the last four digits of your mobile number. When prompted, create a password which can be any 4 to 7 digit code. (A good password is one that’s easy for you to remember, but hard for others to guess) When prompted, record your greeting and name. Your voicemail is now ready to use. Accessing your Voicemail Call 123 to connect to your voicemail Press 1 to listen to unread messages If you wish to replay a message, press 1 Press # to skip to the next message Press 7 to delete a message, or press 9 to save for the next 30 days At any time, press * twice to return to the main menu To change your password, press 5 from the main menu Call *86 from your phone Press 1 to play your messages Press # to skip to the next message Press 7 to delete a message, or press 9 to save To return to the main menu at any time, press * To enable or disable voicemail notifications, from the main menu press 4 for personal options, followed by 1 To change your password, press 4 for personal options, 1 for Administrative Options, then 1 again Checking Voicemail from a Landline Call 1-805-637-7249 Then when prompted, enter your 10-digit mobile number. You can skip the greeting by pressing the * key. When you’re prompted, enter your password. And finally, just follow the menu prompts to check your messages. Call your cell phone from the landline Let it ring till it goes to voicemail As soon as you hear the greeting, press # When you’re prompted, enter your password just follow the menu prompts to check your messages.