What would a living natural language that couldn’t change or evolve look like?

Well, what drives language change? Whatever needs drive language change would not be met by such a language. And speakers of such a language would get very frustrated.

• They’d be bored to death with each other. A major driver is the pursuit for novel and vivid ways of expressing a concept. You would not have them. You would have heard all the ways of expressing excitement a millionfold, and nothing in language would surprise you any more.
• Their facial muscles would be twitching. A major driver is ease of enunciation; that’s how words get slurred together, syllable structures get simplified, phonemes assimilate to each other. That capability would be frozen.
• They would be constantly asking each other “huh?”. A major and contrary driver is ease of comprehension; expressions that become too indistinct, semantically or phonetically, are remodelled so that they can be understood more readily. The easy pathways for doing so would all be blocked off; any attempt to make themselves understood would be trapped in ponderous circumlocution.
• They would yell at each other a lot. Language is a major vehicle of conveying what in-group you belong to, and what out-group you don’t belong to; people unconsciously, and at times consciously, change their language to mark themselves off from others. Without that subtle vehicle, they would have to resort more often to more overt signals of their group identity. Which would probably manifest themselves more aggressively.

Expanding answer promoted by OP.

This was a good, interesting answer. Thanks.

However, there are a few things I’d like to point out/ask, if you don’t mind.

• It seems your answer focused more on the socio-cultural aspect, especially the consequences, while I also had the language itself in mind. That is, the actual phonological and grammatical side of it.
• The various issues you mention are seemingly based on existing languages, while my question was more hypothetical (I didn’t mean an existing natural language).
• For example, you mention ease of enunciation, which means the language would not yet have arrived at the point where speakers feel they no longer need to make it easier.
• Rather than “What if a language suddenly no longer could evolve?”, my question was more about “How would a hypothetical natural language end up in such a state that all or most parts of specch and phonology had become a closed class?”.

An isolating language is going to be crying out for its function words to be reanalysed as affixes, changing it into an agglutinating language. An agglutinating language is going to be crying out for its affixes to be phonologically assimilated, resulting in a fusional language. A fusional language is going to be crying out for its inflections to become phonologically indistinct, and disambiguating function words to be added, resulting in an isolating language.

In other words, there is a cycle of language change, in this and in many other aspects of language; and so long as the forces which bring about language change exist, a language can’t hop off the cycle. The core conflict of ease of production vs ease of comprehension results in language being in an unstable equilibrium: any perturbation (and there are constant perturbations) leads to local language change. And those opposing factors are not intrinsic to any one language structure; there is no language structure that is guaranteed to be stable.

For example, you say “the language would not yet have arrived at the point where speakers feel they no longer need to make it easier.” It never will. A language which speakers feel they no longer need to make easier to pronounce is a language consisting of the single word “uhhhhhhhh”. The contradictory pressure to make the language easier to understand is going to kick in way before then.

What would it take for a language to have most of its parts of speech and phonology be a closed class? Have the language be in a world where there are no novel concepts to express. Failing that, have the language have no compounding, and no phrases consisting of multiple words and a single denotation, which could be reanalysed into a single word, and accented accordingly. It just won’t happen with language as we know it.