What would the minimum number of words needed for a conlang, while still being able to speak with about the same level of information passed as a natural language?
It's really impossible to pick a specific number, like "you need at least 3,427 words to make yourself understandable", without making such claim look ridiculous. The easy rejoinder --- "well, my invented language can make itself perfectly understandable using only 3,426 words: so there!" --- only serves to demonstrate the somewhat absurd nature of the argument.
That said, minimalist invented languages are a thing! (And so are maximal...)
We could look at a couple historical examples of both kinds of language to see what answers our fellow language inventors and natural cultures have come upon:
- Basic English is a good starting point as it's an old project (nearly a century old) and is what the Simple English version of Wikipedia is based on. Basic English seems to get by with 2000 words or so.
- Toki Pona, on the other hand, a philosophical language rather than an international auxiliary language or an artistic language, makes do with about 120 root words. In the article, it's noted that speakers have to resort to combining these root words in order to get their meaning across when they wish to communicate anything that isn't, strictly speaking, minimalist in nature.
- Esperanto has somewhere between 2500 and 5000 basic roots, but again, combining roots allows for greater understandability.
- At the other end of the spectrum, English probably exceeds a million words. And even then, we are as prone to forming compound words as anyone else!
I'd argue that that actual answer to your marquee query is NOT ENOUGH BY HALF.
Obviously, 120 words is far too few to really properly communicate anything but the uttermost basics in the most simplistic terms. 2500 is not enough either, because speakers of languages with so few root words end up relying heavily on affixes and compounds to improve communication. And lastly, I'd argue that even a million words is not nearly enough! Even with a million words, English speakers make use of affixes, compounding, abbreviation and we're sprathingly good at just plain making up words on top of everything else.
It depends on your definition of 'word'.
Tokin pona has 120 word forms, but they are routinely combined into compounds which express more specific information. In Esperanto, as elemtilas said, there are roots which are combined into words — effectively similar to toki pona, only with no spaces in-between:
Esperanto: reĝino (-in- indicates 'female')
toki pona: jan lawa meli ('person' 'lead' 'female')
English has a single morpheme, Esperanto has three morphemes, and so does toki pona. But both Esperanto and English have one word, wheras toki pona has three. So the question of the number of words is not really all that meaningful, as it depends on the structure of your language.
The real question is how many morphemes you need, but even then it's difficult. In toki pona you can combine morphemes to form bigger elements, and each morpheme arguably has a whole range of meanings; from the context you choose the one that is most likely. The word lili means little, small, young, etc. so you cannot know whether jan lili is a young person or a small one. You will typically know from context.
So when thinking about vocabulary size there are several aspects to consider:
- Language structure: do you have self-standing morphemes that are combined to phrases (like toki pona), morphemes that are stuck together in compound words (like Esperanto), or single morphemes that express multiple meanings at once (like English king: 'ruler' and 'male' vs queen: 'ruler' and 'female'). If — like Esperanto — you stick to one basic meaning component per morpheme, then the gender of the ruler would have to be expressed in a different morpheme (or it could be omitted altogether). The same also applies to verbs and tenses, pronouns, etc.
- Specificity: How accurate do you want your language to be? If you want to be able to express exact meanings, then you need different morphemes. In English there are many words such as Earl, Duke, Count, Margrave, etc. which in toki pona would just be jan lawa ('leader'). Toki pona is good for epics, where such distinctions are not relevant, but if you want to write a detailed history of the Norman conquest, then maybe choose a different language.
- Target domain: It also makes a difference what you want to use your language for. If it is an intergalactic lingua franca to be used for trading, then you'd need to be specific about prices and descriptions of objects, but not so much about family relationships. Klingon famously has a vocabulary dominated by words related to fighting and killing, so it might be hard to write a love poem in it (though not impossible). A general purpose language needs to be able to express a wider range of meanings than a domain-specific one.
- Overall complexity of your language: How many cases do you have? How many numbers? Grammatical genders? Any such feature needs to be expressed in some way, which could result in more words. If you have singular/dual/plural, you might need three pronouns I, us-two, and us-more-than-two. But you could choose not to have that at all and simply use us-any-number-from-one-to-many. Hawai'ian has a different set of pronouns to distinguish between us all, including you and us all, but not you. If you have a politeness system based on relative ages, then you might want to distinguish between you-who-is-younger-than-me and a more polite you-who-is-older-than-me
- Aesthetics: you can make do with one word for move-from-A-to-B. And one word for ingest. But any story you write will quickly become very boring, as you will use the same word over and over again. So you want synonyms to introduce some variation (see the remarks about Klingon above). The more synonyms you have, the more words. They're not strictly necessary, but make your language more pleasant to use. In AI, the opposite is actually used to make it easier for computers to handle events: you have a small set of semantic primitives for the basic events (movement, transfer of objects, etc). But you wouldn't want to read such representations for fun.
To summarise: The right number of words for a conlang is how many you need for it to work, given the purpose and structure of the language. It is a bit of an irrelevant question, really, because you won't know that you have enough words until you're happy that your language is usable. And even then, new words are likely to develop over time. In my view, number of words is not a useful metric when developing a language, unless you want to use it as a constraint (as in toki pona).