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
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.