Say you have a Many-Many table between Artists and Fans. When it comes to designing the table, do you design the table like such:
ArtistFans
ArtistFanID
Funny how all answers favor variant 2, so I have to dissent and argue for variant 1 ;)
To answer the question in the title: no, you don't need it. But...
Having an auto-incremental or identity column in every table simplifies your data model so that you know that each of your tables always has a single PK column.
As a consequence, every relation (foreign key) from one table to another always consists of a single column for each table.
Further, if you happen to write some application framework for forms, lists, reports, logging etc you only have to deal with tables with a single PK column, which simplifies the complexity of your framework.
Also, an additional id PK column does not cost you very much in terms of disk space (except for billion-record-plus tables).
Of course, I need to mention one downside: in a grandparent-parent-child relation, child will lose its grandparent information and require a JOIN to retrieve it.