[Virginia-American]

Has anyone ever shown that anything is actually 'irreducibly complex'?

Yes, that is what ID shows - systems that are irreducibly complex

Actually, I was asking for examples, especially of the method of proof.

[Dimensio]

Actually, I was asking for examples, especially of the method of proof.

My experience has been that the "examples" are simply handwaving and appeals to ignorance and incredulity: they can't imagine how some organism or organ could have evolved, therefore it didn't.

[gore3000]

Actually, I was asking for examples, especially of the method of proof.

The bacterial flagellum for one. You can see an explanation of it here:  Bacterial Flagellum.

The method of proof is by showing that the different parts of the structure are so specific and so interconnected that they would be useless for anything except working with the other parts. The idea for intelligent design came about from the discussions pro and against evolution in the 19th century. Specifically from Darwin's quote below:

If it could be demonstrated that any complex organ existed, which could not possibly have been formed by numerous, successive, slight modifications, my theory would absolutely break down. "
Charles Darwin, "On the Origin of Species by Means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life", Chapter 6.