“Can't you make an exception just this once?”

You and your friends are trying to get into a nightclub, but the club has a dress code that doesn't allow people wearing shorts to enter. Your friend is wearing shorts, so the bouncer won't let him in. You beg the bouncer:

Can't you make an exception just this once?

can't you (do something)

You can ask "Can't you ___?" instead of "Can you ___?" when you're not just asking a question, but trying to convince someone to do something. In the example above, you're trying to convince the bouncer to let your friend in. In the following example, you're trying to convince your son, who's playing a noisy game, to play in another room so that it doesn't distract you:

Can't you do that in the other room?

The tone of the first example, "Can't you make an exception...?" is polite. The tone of the second is annoyed. But both of them are used to try to convince the listener.

make an exception

When you're the person who makes or enforces a rule, people will sometimes ask you to let them break the rule. If you let them do this, you are "making an exception" for that person. For example, if a teacher has a rule that homework has to be submitted by a certain date, he can "make an exception" for a certain student whose computer broke. Use "for ___" to indicate the person who is allowed to break the rule:

I don't usually give my number out to people I've just met, but for you I'll make an exception.

(do something) just this once

When you're trying to convince someone to do something that they don't want to do, one way to persuade them is to say that it's "just this once". It means that they only have to do it one time. In the future, they won't have to do it again. People use this phrase a lot when asking for favors from people. Like when you want to borrow money from a family member:

Can you loan me a couple hundred bucks? Please, just this once.