Q:

Is there a linguistics equivalent to Turing completeness?

In computer science, programming languages can be described in terms of "Turing completeness", basically, whether a programming language is capable of expressing any* algorithm. A non-Turing-complete programming language has notable limitations in terms of what ideas it can express.
Does such an idea also exist for spoken languages, as some measure of empirically describing the limitations of the ideas a language can be used to express?
My first thought as to a way of answering this question was to try to think of an idea that couldn't be expressed in a given language, quite the feat given the thinking itself happens largely within the confines of that language. I stumbled on ideas related to alternative versions of reality, for example, Had he not gone back in time to change the past, he wouldn't have eventually had to avoid his younger self in the future. English seems to have the tenses to cover such a situation, but it's not difficult to imagine a language that doesn't.
*This is an oversimplification of course. More accurately, a Turing-complete language can be used to express a Turing machine, which in itself can solve most any problem, but caveats abound

A:

In the realm of natural language, the "ideas a language can be used to express" are basically "any": all languages are capable of expressing any idea, so there's only one category of expressive type. Languages do differ in the way that they express a given idea. Assume a language Gwambomambo which lacks the word "recursion". That very word could be introduced into the language, just as "ballet" or "ghee" was introduced into English; or, a word might be invented using traditional roots of the language (e.g. "thing sits on itself"). Some languages have specific tenses for negative propositions and some use words like "not" to convey the idea; some languages have different forms of words to indicate that there is just one, or many, or maybe even just two, of the thing in question – other languages don't do this but do allow you to say "1 child", "many children", "more than 1 child" etc.
The greatest disparity between languages is lexical differences, that is the fact that we need a long expression to convey the notion "2 year old male reindeer", whereas languages spoken by reindeer-herding cultures usually have a single word for this. 

A:

In computer science, one essential property of all Turing-complete languages is that they are able to describe, "in their own way", how they themselves work.
For example, you can use a Turing machine to express how a Turing machine works.
Similarly, you can write, for example, a Prolog program that can interpret Prolog programs.
In the linguistic realm, it would seem to me that an analogon of "Turing completeness" is a language that can express concepts of the language itself.
For example, you can use English to describe English grammar, English words etc. Similarly, you can use Latin to describe Latin grammar. But you will in all likelihood not be able to use a language consisting of, say, 3 different whistled tones to describe how the tones themselves are constructed and related. This follows from the simple fact that the vocabulary and grammar are too restricted to talk about anything on the meta-level, or at least under reasonable assumptions (note that you can encode any information even with a single whistled tone, by reasoning about the time intervals between different occurrences of the tone). 
In computer science, a Turing machine that describes, in general terms, how a Turing machine works, is called a universal Turing machine. Characteristically, a universal Turing machine is able to execute its own description, arbitrarily deeply layered.
So, as a linguistic analogy, a language that can be used to express properties of its own constructs could maybe be called a universal language?
In any case, I think this reflective ability is a good test to see whether a language, any language, is expressive enough also for other important tasks.

A:

I have had the same thought before and this is what I have found. 
There are two main concerns. Semantic completeness and grammatical completeness.
Semantics: A language needs a minimum set of meanings, but the lack of some random arbitrary noun representing an abstract or concrete thing does not make the language incomplete in terms of thought. If the language lacks such a word, it's likely due to the thing being unknown to the speakers of that language. A noun is interchangeable in the structure of a language so you can simply add a noun for it and voila they can think about it. To have the thought of a cup, you must know of cups, but that is not a limit of your language. We are only concerned with the ability have thoughts, ie. fundamental semantic structures.
For semantic completeness, I refer to the Semantic Meta Language. There are 62 primes. For instance, the word 'kill' can be defined in these primes as follows.
An explication is a breakdown of a non-prime concept into prime ones.

Someone X killed someone Y:
  someone X did something to someone else Y
  because of this, something happened to Y at the same time
  because of this, something happened to Y's body
  because of this, after this Y was not living anymore

Source: https://en.wikipedia.org/wiki/Natural_semantic_metalanguage
I also think Cognitive Linguistics might have insight here. A language surely must have ways to express all the image schemas, for example as containment "in".
Grammar: I do not know yet, but I imagine there is a list somewhere. If it hasn't been constructed, it really ought to be! For instance, a language must have a way to determine which word in a sentence is the subject and which word is the object.
Clearly there is a minimum set, or else creoles would not differ so much in their sophistication from their parent pidgins.
Finally I want to add that though I mentioned nouns and verbs need not be there for 'thought completeness', perhaps they do for a sort of minimum human completeness. There are certain concepts which are found in all languages because all people deal with them  no matter how different they are. Words for basic things like 'people', 'animal', emotions, basic colours, basic objects and animals 'tree', 'bird', basic verbs like 'run' and 'sleep'. Sky. Ground. For a good example of such words, you can check out the Swadesh list. However, the Swadesh list is not exhaustive, just a selection. A more exhaustive set of such concepts could be constructed.

