Are there any languages where adjectives have special endings which depend not on the gender of the noun they modify, but on the order they are placed in the sentence?
For example, say the first adjective always gets a “t”, the second always gets an “m”, and the third always gets a “ch”.
**A bluet shirt.
An expensivet shirt.
An expensivet bluem shirt.
A bluet expensivem shirt.
A bluet, expensivem, cottenech shirt.**
You just invented one.
Never heard of anything like that.
In English, when you use multiple adjectives to modify one noun, there are some rules (which I think are more “rules-of-thumb” than formal rules, AFAIK) about whether the adjectives are separated by commas.
I don’t see why a language would evolve that rule. Adjectives that agree in gender or case with nouns do so for clarity, so that if they are separated in a sentence, you still know which adjective refers to which noun. English has pretty strict syntax, because it doesn’t have cases or genders, and doesn’t modify adjectives (other than that/these, and soforth) to agree with plural nouns. Unless there’s ever a need to refer back to one of the adjectives, and not the others, with some kind of pronoun of place in order, I can’t think of a reason to modify the words for place in order. You can see where they are.
I’m with Rivkah–because I can’t imagine how that marking would have any semantic significance, I don’t see where a language would develop it. And such endings, though they might encode syntactic info, would just be repeating directly the info given in ordering.
INterestingly, some adjectives subcategorize for relative order. Don’t know why, but consider:
big, old, ugly building
*ugly, big, old building
I could see how something like this (although not necessarily limited to adjectives) would be potentially useful if it helped to disambiguate embedding, in such scenarios as The rat the cat the dog bit chased escaped.
If that became (for example): The ratn the catm the dogl bitel chasem escapen, it would be easy to see which parts belong to each other (OK, it makes no sense now because those aren’t proper words).
But it’s probably not a common enough issue, or a great enough utility, for something to arise to solve this non-problem.
I’ve heard a tale of a ruler consulting the Oracle of Delphi about whether to go to war, and getting an answer that was cryptic in this way: It could be parsed as either “You will go, you will return, you will never die”, or as “You will go, you will never return, you will die”. A word-ending rule that can remove that sort of ambiguity would be useful… but the rule proposed by the OP wouldn’t do that.
In any living language, all rules are rules of thumb first and foremost; linguists may or may not come along later to observe, figure out, and write down what they suspect the formal rules are, but the informal rules of thumb are primary, because those are what the actual speakers use. That’s the heart of descriptivism.
Later on, after the descriptivists have done the actual work, prescriptivists come along and say that their rules, based on a mix of misunderstood linguistics and base prejudice, are the only True And Correct Rules and everyone who doesn’t conform is ill-educated and ill-bred. We will pass lightly over the likely race and class differences between the prescriptivists and those the prescriptivists consider to speak a debased and corrupted dialect…
In French, past participles, when used with the verb avoir, will take the gender of the object, but only if they appear after it in the sentence:
J’ai appelé une voiture pour nous emmener à l’aéroport.
La voiture que j’ai appelée nous emmènera à l’aéroport.
But this is more about convenience of parsing.
The wording was:
You will return not die in war.
A simple little comma makes all the difference.
You will return, not die in war 
You will return not, die in war. 
Either way it goes, the sly oracle has an out.
If there was such rule, the oracle would probably fastidiously avoid invoking it. Prophets who make unambiguous claims tend too soon find themselves proven wrong.
This was discussed on Wikipedia in the context of adjective order in English.