Why is "our today's meeting" wrong?

This seems baffling, but what is special about today's?

I think it comes down to this:

We cannot use two genitives to modify a single noun.
At least not outside Indian English.

Today's is a "genitive".
_{I don't want to use the common possessive here, because it's hard to imagine actual possession in this case. For this answer I will use "genitive" to refer to the form that is used to indicate possession and that was once called genitive.}

Looking for other examples that sound plain wrong, I noticed that it seems impossible to have two (or more) "genitives" that relate to the same noun unless it is possessive, and we actually intend to convey shared possession. In that case we still form a single genitive:

John and Paul's car.

Now, if we use any other noun or adjective to modify our noun, it always follows the "genitive:"

John and Paul's red car.

Note also that when we have a "genitive", we do not use an article. With non-genitive modifiers, we usually have to use an article:

An old newspaper.
John's newspaper.
John's old newspaper.

Note that "old John's newspaper" is valid, but means something completely different!

In our yesterday's meeting, we have two "genitives", namely our and yesterday's, but only one noun, meeting. For most speakers of English, this causes a clash, either grammatical or semantic, meaning that the sentence sounds wrong.

The same would happen with that car:

*Our John's car.
*Our your car.

_{Note that our John's car can be parsed fine if we assume that the car belongs to our John. In that case, our does not modify car, but John. See also a bit further down, where I discuss John's sister's friend.}

We have no problem with the addition of non-genitive modifiers in between a single "genitive" and the noun:

Our great old fast red car.

As Tim Romano mentions, we can have a double genitive like this:

John's sister's friend.

Here, friend is modified by John's sister's, acting as a single genitive. John's does not modify friend, it modifies sister. We can see this because we can add modifiers in between the two, and they will also modify sister, not friend:

John's younger sister's friend. -> the sister is younger
John's sister's younger friend. -> the friend is younger

As Janus Bahs Jacquet points out, multiple genitival constructions are usually parsed as nested, contrary to multiple adjectival constructions, which can be parsed parallel, all referring to the same noun(phrase).

X's Y's Z -> [X's Y]'s Z -> Z of [X's Y]
John's brother's wife -> the wife of [John's brother]
Alice's friend's phone number -> the phone number of [Alice's friend]

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Note that I mentioned most speakers of English. This may become untrue quite quickly, because it seems that in the fastest-growing dialect of English, Indian English, this double genitive is not frowned upon, at least not always. The phrase our today's meeting is commonly used in Indian English, even though other dialects of English frown upon it.

The mentioned examples in the comments of our today's specials and our today's speaker will, I think, sound off to many speakers, but possibly not as much as our today's meeting.

It is entirely possible that a weakening sense of possession in the case of today's will make such double "genitives" slowly more and more acceptable for a growing group of speakers.

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And then there is a slightly broader way to look at this, and to take in what I noticed before about the absence of articles when we have a "genitive":

As Janus Bahs Jacquet notes (and I am more quoting than paraphrasing here):

today(’s) acts as a deictic. Deictics always add definiteness to a noun phrase, and so do possessive pronouns and determiners.
You can’t mark a noun phrase for definiteness twice (or mark for both definiteness and indefiniteness).

That’s why neither “the/an our meeting”, “the/a today’s meeting”, nor “our today’s meeting” works: today’s makes it definite, so you can’t add another (in)definitiser.

I agree with oerkelens' answer, but I am surprised no one has mentioned that the expected form, at least in British English, would normally be "our meeting today". For example, "We welcome Professor David Morrison to our meeting today." While I think most of the grammatical arguments are valid, the main reason I sense "our today's meeting" to be wrong is that a native speaker wouldn't say it: the correct idiom is "our meeting today", at least in most contexts.

Simple: you don't own today.

The normal phrase is "our meeting today".

However, note: "All our yesterdays" is legitimate, but poetical and I can imagine a similarly flowery use of "our today" or even "our todays" but stretching that even further to have that today then possess the meeting just doesn't look like it would ever work to me.

I think I have an example in which the phrase "our today's meeting" might be uttered by a speaker of English, at least in informal conversation.

Alice and Bob are in an office in New York, USA, talking on the phone to Colleen, who is in an office in Perth, Australia. Alice, Bob, and Colleen are members of a team working on a project together, for which they have a status meeting (by teleconference) every weekday at 7 am, New York time. The conversation below, however, is not from the regular status meeting; it occurs when the local time in New York is 7 pm on Tuesday, but the local time in Perth is 7 am on Wednesday.

Alice: "Doris made a good point in today's status meeting." [She is referring to the meeting that occurred at 7 am on Tuesday, New York time, which (for Bob and Alice) is the same day as the day of this conversation.]

Colleen: "What do you mean? Today's meeting hasn't even happened yet." [She is referring to the meeting that will occur at 7 pm on Wednesday, Perth time, which (for Colleen) is the same day as the day of this conversation.]

Alice: "Sorry, I meant our today's meeting, not your today's meeting."

I would hope this phrase would not show up in the team's final report, however. It's extremely awkward in print.

I'm by no means a language expert, but when I look at "our today's meeting", I feel it's wrong because of where time-qualifying words should go. In English, they always go after the verb (I went to the shop yesterday), while in my native German, and other Germanic languages (and probably other languages too) they come before the verb (I went yesterday to the shop...). I think you call that rule "time-manner-place". Here is a useful link! http://en.wikipedia.org/wiki/Time%E2%80%93manner%E2%80%93place

As an alumnus of the Haberdashers' Aske's School, I say with some certainty, there is no rule that you can't have two possessives before a noun.

But why is this OK and "our today's meeting" not? In this case, the possessives qualify in a chain - Robert Aske was a freeman of the Worshipful Company of Haberdashers, thus "Haberdashers' Aske", and the school was named for his bequest, thus {Haberdashers' Aske}'s School.

In "our today's meeting", both possessives qualify "meeting" and this is what is not OK.

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Of course, it's trivial to make a chain of possessives. Bob's mother's dog's puppy's teeth have come through.