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]
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.
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.
