This is an interesting question. On the one hand, I think that you could argue that there is indeed some limited subset of words that can be used to say everything that can be said in a given language. One way to prove this, perhaps, would be to count all of the words that are used in the definitions in a given dictionary. I suspect that in most dictionaries, the total number of unique words used in the dictionary definitions is less than the total number of defined words.
This is related to what you wrote, "in practice, words are continually added to a language when new things need to be described". Consider new technologies for which new words are invented: these can generally be defined using the existing set of words in a language. The new word just provides a one- or two-word short form method of referring to that technology.
On the other hand, there are certain things - and I'm thinking specifically of human experiences as one set of those things - which are not easily summarized in a sentence and these get their own words. Consider emotions as one example. As the set of "known" human experiences (emotions, perspectives, etc.) grows, based on a more sophisticated understanding of human nature (via psychology, literature, etc.), new words are needed. Sometimes these words are borrowed from other languages, like "schadenfreude" and "empathy" (I was surprised to read that "empathy" first appeared in English in 1909, based on another German word).
On this basis, it would only be possible to limit the set of possible words if it were possible to limit the set of human experiences, which I don't think is possible. A transhuman/sci-fi example would be if a method was invented to perceive visual wavelengths that are currently outside the human visual spectrum as colours (i.e. infrared and ultraviolet, but perceived directly in the same way we perceive red and purple). I can't currently imagine a way (although perhaps there is one) of describing something like this using the existing set of words.
This is related to what you wrote, "in practice, words are continually added to a language when new things need to be described". Consider new technologies for which new words are invented: these can generally be defined using the existing set of words in a language. The new word just provides a one- or two-word short form method of referring to that technology.
On the other hand, there are certain things - and I'm thinking specifically of human experiences as one set of those things - which are not easily summarized in a sentence and these get their own words. Consider emotions as one example. As the set of "known" human experiences (emotions, perspectives, etc.) grows, based on a more sophisticated understanding of human nature (via psychology, literature, etc.), new words are needed. Sometimes these words are borrowed from other languages, like "schadenfreude" and "empathy" (I was surprised to read that "empathy" first appeared in English in 1909, based on another German word).
On this basis, it would only be possible to limit the set of possible words if it were possible to limit the set of human experiences, which I don't think is possible. A transhuman/sci-fi example would be if a method was invented to perceive visual wavelengths that are currently outside the human visual spectrum as colours (i.e. infrared and ultraviolet, but perceived directly in the same way we perceive red and purple). I can't currently imagine a way (although perhaps there is one) of describing something like this using the existing set of words.