You could say... " John had had a knee operation in 1995" or "the first car I had, had a bald tire" etc that sort of scenario...

I can't think of any, it should be "has had" or "have had", I think :-)

"Had had is the past perfect form of have when it is used as a main verb to describe our experiences and actions. We use the past perfect when we are talking about the past and want to refer back to an earlier past time[...]. In these examples, note the use of before, after, already and by the time as a trigger for the past perfect. Note also that the contracted form of had had is 'd had. She'd had a lot to drink and wasn't capable of walking home by herself. After he'd had a good night's sleep, he felt much better. She sacked him before he had had a chance to explain his behaviour. By the time he was twenty he'd already had four different jobs. I'd already had a word with Joan about re-locating to Manchester and now she's had time to think about it, she quite likes the idea. Note that past perfect forms are a feature of if-clauses in the third type of conditional sentence when we are explaining past actions or regretting past inaction. Thus, had had is likely to appear in this construction: If I hadn't had a good education, I would never have got this job. If she had had children later in life, she would have been a better mother. If I'd had another ten minutes, I would've finished the examination paper. Had they had any savings they didn't need, they would've re-paid their son's student loan." Source and further information: http://www.bbc.co.uk/worldservice/learningenglish/grammar/learnit/learnitv343.shtml

You can say Maria had had a pregnancy test before she gave birth to a baby girl called LK.

For example, 'I had had some difficulties with (a task) in the past, but now I find it easier.'