Will it ever happen that we design a table that doesn\'t need a primary key?
Primary key not mandatory but it is not a good practice to create tables without primary key. DBMS creates auto-index on PK, but you can make a column unique and index it, e.g. user_name column in users table are usually made unique and indexed, so you may choose to skip PK here. But it is still a bad idea because PK can be used as foreign key for referential integrity.
In general, you should almost always have PK in a table unless you have very strong reason to justify not having a PK.
Link tables (in many to many relationship) may not have a primary key. But, I personally like to have PK in those tables as well.