The Clarify database is an example of a commercial database that has no primary or foreign keys.
http://www.geekinterview.com/question_details/18869
The funny thing is, the technical documentation goes to great lengths to explain how tables are related, what columns to use to join them etc.
In other words, they could have joined the tables with explicit declarations (DRI) but they chose not to.
Consequently, the Clarify database is full of inconsistencies and it underperforms.
But I suppose it made the developers job easier, not having to write code to deal with referential integrity such as checking for related rows before deleting, adding.
And that, I think, is the main benefit of not having foreign key constraints in a relational database. It makes it easier to develop, at least that is from a devil-may-care point of view.