The Past Perfect is unavoidable in some situations where it's important to show that an event or action took place before some other event or action also in the past.
A, When I arrived at the theatre, the play began.
B, When I arrived at the theatre, the play had begun.
In A, the play began AS I arrived.
In B, the play was ALREADY in progress. It began before I arrived.
This is an important difference.
However in the context of 'had had', it may sound inelegant but it's absolutely standard. It's the identical construction as in French, German and Italian (as shown by @Pontus), the only difference being that in those languages, the form of the Past Participle is not the same as the auxiliary verb 'had'.
In the examples given by @Pontus, they must all be translated into English as 'had had'.