I have had had a question

You can say, "I have a question."

You can say, "I have had a question for a long time." Right?

But what about, "I have had had a question ..."? Optional side question: how do you punctuate the previous sentence? Is "have had had" grammatically correct? If so, what does it mean? Also, is it then correct to use "have had had had had had had," and does each use with a different number of hads mean something different? Do these tenses have a name? Would you only use these tenses in unique time travel scenarios?

"I have had had a question," sounds weird but not necessarily wrong. "Having had had a question, I asked it," sounds okay. Or something like "Having had been a person..."

EDIT

So it seems that "Having" in "having had had" does not necessarily have anything to do with the tense? It still seems grammatically correct and not completely unusual to say "having had had" or even "having have had" for that matter.

Bob: I had had a question, but I never got a chance to ask it. 
Bill: Having had had a question that was never asked, I understand how you feel. 

I also found a few (very few) examples of "having have" by googling. I don't know if they're grammatically correct, but here's some:

"24.6% reported having have been injured due to excessive consumption of alcohol."

"When I got up to clear the table, only having have eaten a third of the fish—which was shockingly filling—I noticed that..."

If these uses of "having have" are okay, then doesn't that mean that "having" at the beginning of such a sentence is different than "have" when used in the present perfect.

But sometimes, doesn't "Having" in similar-looking sentences have something to do with tense?

I have a question.

Present tense. Right now I have a question.

I have had a question

Perfect tense. At one point, I had a question. It may or may not be true that I still have it now (might be made clear from the rest of the sentence).

I had had a question.

Pluperfect. At one point it was true that at an earlier point it was true, that I had a question. "I had had a question for some time, but I never got a chance to ask it".

Now, in each of these the verb have is used once in the sense of "to possess" or "to hold" and in each of the second two it is used as an auxiliary to modify that other have.

You propose.

?I have had had a question.

Which we would presumably have to interpret as some sort of super-perfect stating that at some point it was true that at an earlier point it was true that at an earlier point it was true, that you had a question. It's not clear what you are saying about the current state.

As such, you've just created a non-standard variant of either the perfect or the pluperfect, but you leave us confused as to which - and imply that you are confused about the matter yourself.

You suggest it might be better as:

?Having had had a question, I asked it.

But this presumably would mean that it being true that at one point it was true that at one point you had a question. Again, it's not clear just what this is supposed to mean, and one possible interpretation has this as impossible (because one way of untying the knots leaves us with the suggestion that you no longer had a question at the time you said it).

In all, this reminds me of some comic nonsense writing that has been done - sometimes well - but were the whole point of it is that it was not good English. Barring that goal, none of this makes any sense.

You talk about "unique time travel situations", and I could see someone deliberately engaging in this sort of abuse of auxiliaries to describe that. Still, the implication is "this time travel has so messed with the logic of causality that English grammar can no longer work to express the resultant mess". Once you're doing that then you've deliberately thrown the rules of grammar away for effect anyway, so asking if it's grammatical is not just besides the point, but counter to it.

Really though, this is not grammatical English. Nor is it a useful construct to anyone who perceives time and causality as being related things.

I have had had doesn't exist in English as a verb form.

It's a clunky way of expressing something that could be better said in other ways, but it's still a valid expression.

Let's say you're recalling a story about someone asking a question in a lecture you attended 2 years ago. "Having had a similar question myself, I could sympathize." This doesn't make it clear when the past tense that "had" points to was.

Maybe you'd been thinking of asking the same question at that time, or maybe you'd ended up having that same question at some point before then, but no longer did by the time the lecture came around. But either way, you no longer have it now, or you'd say "I've had..."

But if you put it differently, "Having had had a similar question myself, I could sympathize."

If you put it like this, it becomes more clear that the time you had the question was before the main incident (the lecture) that you're talking about.

So there is a justification for this use, I think, but really, the "having" here is just used as a replacement for a "because," so it could just be replaced with "Because...I" or "I...So," like : "I had had a similar question myself, so I could sympathize." "Because I had had a similar question myself, I could sympathize."

So the only way that "having had had" works is if there are multiple different meanings of "having/have/has/had" combining.

So now you have had my two cents on it.