Is there a name for this manga-anime mouth expression? (`3`) Is there a name for this expression?

They do it a lot in YuruYuri, too.
This is a pout, and is based on the real life expression of the same name. 

While it can be used as way of making oneself look more attractive, it's more often used (especially in anime) as a childish way of expressing displeasure or annoyance, in a similar way to the puffed out cheeks expression.

See here and here for additional definitions.

The "3 face pout" is an exaggerated form of a pout, and there are other less extreme cases to be found elsewhere. Some examples of varying intensity are included below.





Additionally, the "3-face" is not limited to only a pout. As a shorthand form of displaying puckered lips, it can also be seen upon characters either awaiting or giving a kiss, or on those whistling.