Is it grammatically correct to say, "I would have had to have had....."?

Context is all. 'I would have had to have had a key to unlock the door when I arrived last week' is good English. What Glee and Jellikit told you is, I'm afraid, not. (You could shorten it by saying 'I'd have had to have had...' but that's all.)

The first 'I would have had' is about need at a point in time; the second is about possession. So your words can be rewritten as: 'I'd have needed to have possessed a key...' If you made it, 'I would have needed to have a key' you are saying the continuing possession of a key would have been necessary. It wasn't. One turn and you'd have been in. ANF's suggestion is very good, provided whether or not you continued to have a key matters. If it doesn't, then say, 'I'd have needed a key.' Context is all, provided you remember that meaning forms a large part of context!.

I don't know your age, but I suspect that you've been let down by your teachers, like almost everyone under the age of fifty in the US and the UK. Political correctness has led to a generation of teachers who themselves weren't taught how to speak and to write English properly.

If you think about that, you'll see that the situation is hopeless. We are a species in intellectual decline.

Better still if you had said.

I would have needed.

Your sentence and the alternatives given are just a bit messy. Why use all those words when one would suffice.

Too many tenses. You can say "I would have had to have" or you can say "I would have to have had" depending on where you want to put the infinitive and where to put the past tense. One past tense is enough.

To be honest I think it depends on the context you are using, so I can't really critique you on your grammar.

yes, but to make it less confusing you can say

i would have to of had...

You can leave off the first had....

"I would have to have had..." =)