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"'s" indicates possesive case. Although it's absurd to say that "today" owns "weather", possesive case can indicate other relationships too. For example, in "Picasso's paintings" it means "by Picasso" and in "children's stories" it means "for children". So "today's weather" can be interpreted as "weather for today". So all is good.

BUT there's another seemingly valid construction, namely "I want to know today weather", where "today" is a noun. We could say that in said phrase "today" is an attributive noun (A.K.A. a noun adjunct), thus implying that it means "weather for today". (Like in "dog food" word "dog" is an attributive noun that makes meaning "food for dogs". In other words, an attributive noun can indicate "for".)

Is there anything wrong with my reasoning? I was told that "Today weather" is incorrect.

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Well, I think the problem with your reasoning is that "today" is not a standard noun.

We can use standard nouns in this construction, such as in "I want to know the mountain weather" or "I want to know the aviation weather"—both of these sentences are okay (though they sound just a little strange).

If "today" were a standard noun, then we would be able to say "I want to know the today weather". But it's not, so we can't.

Jason Orendorff's answer to “Is ‘yesterday’ a noun, an adjective or an adverb?” states that according to The Cambridge Grammar of the English Language, the word "today" is actually a pronoun, not a noun. And pronouns cannot be used like attributive nouns.

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