Is it possible to send a letter through the mail without having to add the return address or is it required?

A return address is not, strictly speaking, required for the USPS to deliver mail. The return address is used if the mail is undeliverable for some reason; if it is missing, the letter will be treated as a "dead letter". This may mean that the post office might open it and search it for clues as to where it should be returned to, so if there's anything in there you'd be embarrassed by, just put on the return address already!