Don't immediately dismiss this! Just give me a moment to explain and you might realize it's an idea that might have merit, especially considering the question title is a little misleading. Also, I'm no conlang or linguistics expert, so if there's a couple small things I got wrong but don't necessarily make this impossible, especially if fixing it might make it possible, correct me in the comments and I'll integrate it into my question to make the question something that can actually be answered.
Could you construct a language--or, at least, a grammar system--with only three words? OK, so first of all, "word" is not the proper term for what I'm trying to say, but I don't know of a term that fits. So. When I say three words, I mean three combinations of consonants and vowels that morphemes can be derived from. From now on, though, I'm going to refer to them as "words", just because it's the only term that vaguely fits. Again, don't dismiss me immediately. Let me explain my idea, and then please critique it.
The three base words would each have particular consonants and one particular vowel, in the structure 1v2v3 (where v represents the vowel, and the numbers represent the three consonants). For a silly example, let's say the three words are:
Sazar: Fire (s, z, r; a)
Kitiv: Lie (k, t, v: i)
Lomon: Truth (l, m, n; o)
(If I were to actually construct this language, I would definitely use different base words, but let's stick with these for the sake of example.)
There are thirteen distinct combinations of these letters to be derived from these words (1, 2, 3, v, 1v, 2v, v2, v3, 1v2, 2v3, v2v, 1v2v, v2v3), and that's without rearranging the letters at all. If you allow for rearranging letters, there are thirty-one distinct combinations (if I did my math right). Including the original "word", it's thirty-two.
All these combinations templates can result in 96 different roots/suffixes/prefixes/morphemes/etc. If you somehow made each combination represent some kind of modifier that can be made to a root word, could you construct, if not a language, then at least a grammar system, this way?
For example, if we made v3 represent plurality (-iv=singular, -on=plural, -ar=dual) and v2 represent person (-it=first person, -om=third person, and -az=second person) then that could be the conjugation for the language, giving you:
We: -iton (or -itar for special, dual cases)
You pl.: -azon (or -azar for special, dual cases)
They: -omon (or -omar for special, dual cases)
Then, -omiv could be extended using v1, which would represent gender (-as=neuter, -ik=female, -ol=male). Using this, the conjugation for he would be the way-too-long-for-conjugation "-omivol".
You could make a modifier for roots that determine whether it's a noun, verb, or adjective, and you could also have the three tenses (it's fascinating how much language fits into groups of three) and make cases using combinations of the vowels and consonants, and you would have to stretch some things for prepositions, but maybe instead of having prepositions be words, you could make this an entirely written language and use visuals to convey prepositions (essentially making sentence order influence the physical positions of the objects the words refer to, instead of the grammatical meaning of the words).
I would probably need to create a couple (or many) other roots that the modifiers would be added to (or at least start with more flexible words than fire, lie, and truth), but could I at least base the grammar system off of only the three original words and modifiers derived from those three words?
I don't see any obvious issues with this language, besides long words, but I would appreciate it if more experienced conlangers could critique this. Also, if there are no issues with a conlang with only the grammar based on the three base "words", what about an entire language, where you could only build basic words by combining roots based on the three base words? This would limit the number of morphemes you could use for grammar, which might be an issue if there are a certain grammatical components every language needs and there aren't enough morphemes for all the components.