The use of the word the past participle -- "had" as in: "had had."

NOTE: I hope you don't mind the little bit of humor I used in the example short paragraph below. As the creative writer that I am, I've often found that words and sentences, sometimes have a habit of writing themselves into their own story, a story that the author did not originally intend. This was the case here.

This is a three part question.

I've often read and heard the past participle word “had” used in the following way:

“had had”*, but it's usage always struck me so very strange. I mean, when seen in a sentence it looks pretty funny, the way it's “doubled-up”, but it sounds pretty good, when heard at loud.

1) Is there a grammatical rule for this unusual usage?
2) Are there any other past participles, or – for that matter – any other word(s) that is “doubled-up” like this?
3) Does this method of “doubling-up”words like, e.g. “had had” have a name?

Now, just to set the scene: An old hillbilly attends his very first live opera in the big city. Afterward he complains to his wife:

“Honey, after just thirty minutes of hearing some fancy dressed lady screech and wail a bunch of  strange sorry sounds, I *had had enough of all her caterwauling and high-tailed it out of that big place, as fast as my two feet would take me.”