Could you have a language with only three words?

Your terminology is a little strange, but it sounds like your suggesting something like the tri-consonantal roots of Arabic. For example, د ر س represent the idea of acquiring or giving knowledge. يدرس = to Study; يدرّس = to Teach. The only difference between these and the root is the adding of ي (an infinitive marker) to the start of the word and a ّ to 'to Teach' (which acts as doubling the ر). Vowels and subvowels go between the letters to change the meaning of the words.

Is that similar to what you're talking about or am I off the mark?

From an information theory point of view, you can express anything if you have at least two signs: computers do this through the use of binary, where the signal is either "on" or "off", or "1" or "0". If you have a "message" with just one bit of information, you can express two meanings, which are "on" or "off". If you want to express more complex ideas, you need a longer message. For example, if you want to express the decimal numbers from 0 to 15 in binary, you need at least four bits to represent them.

We as people have created a system where we know beforehand what the bits mean when we send and receive them. So if we send a message consisting of four binary digits, e.g. "0010", then we know that that means the number "2", because that's the convention we've agreed upon.

So you could certainly expand this principle by using three signs instead of two, and use conventions and codes to create meanings. The snag is that the complexity of the message would increase massively. For example, if I want to write the word "message" in English, I only need a total of seven symbols, and each symbol can be one of 26 signs (the English alphabet.) If I want to write the word "message" in binary, however, it will look like this:

0110110101100101011100110111001101110011011000010110011101100101

Which is a considerably longer message, because I only have two signs to work with, not 26.

The other disadvantage is: you need a convention or a "code" which both the sender of the message (the speaker) and the receiver (the listener) to be understood before a message can be transmitted. If you saw the above binary sequence out of context, you would have no way of knowing it meant "message", because I haven't told you that it represents a sequence of ASCII characters. It could just be meaningless computer code, or a very large number, or part of a recipe for a salmon bake.

So you could theoretically create a language with three "signs" and combine them to make meanings. For example, if your words were "ka", "pi" and "ol", then you could have the following sequences:

kapi = woman
kapiol = man
olka = cat
olpipika = dog

And so on. But you would have two problems with such a language:

1. The complexity of each resulting "word" would vastly increase as you make the message longer, and

2. There's no obvious relation between your signs and your word, so you would have to make your messages understood by pre-existing codes, agreed between the speaker and listener.

I haven't (and won't) check your math, but you have defined a grand total of 96... I'll call them blocks. I'll take the position that a block represents a sufficiently new type of language element that it doesn't have an existing name already. Maybe they're a special case of morpheme or of something else, I don't really care. They're blocks, and there are 96 of them.

You allow for 96^2 and 96^3 combos, which is nearly 900,000 blocks or polyblocks. That's plenty of room to define a language. That's bigger than many lexicons, and you can certainly spare a large subset for grammatical functions. Chinese for example is said to have 50,000 characters --- yes those are graphemes and there's apples and oranges here, but you take the point.

Imagine a person who only speaks and writes Chinese and has no awareness that other languages even exist. One day they are told, you can write everything in a language using 26 letters and some punctuation and so forth, you can get away with old-school ASCII with a maximum of 256 graphemes! He'd think you're talking doo-doo kaka.

You could probably do it with a great deal less.

After seeing a lot of different things thrown out there, I've decided to take a shot at this one. I think what you are suggesting is totally possible and plausible. The confusion of answers arises from that fact that a "word" can be defined in a couple different ways.

One reputable definition which allows for your proposed linguistic system is found in the Collins Dictionary:

A word is a single unit of language that can be represented in writing or speech. In English, a word has a space on either side of it when it is written.

By that definition, you have designed three possible "units" which can be assigned the title "words" legitimately, and your language system works.

On the other hand, you have other possible definitions, such as that found in Meriam Webster:

A speech sound or series of speech sounds that symbolizes and communicates a meaning usually without being divisible into smaller units capable of independent use

If you opt for that definition (which some people do by default), it's a bit harder to stretch "word" to define what you're talking about. One of your proposed "words" communicates no inherent meaning, and is necessarily divisible into smaller units. Of course, it's a bit ambiguous, and one could argue that this definition of "word" obfuscates the nature of compound nouns etc.

In short, I would answer, "It depends." If you define a word simply as a unit of speech separated by spaces, you're all set. Other linguists might argue however that you ought to define the different "words" as different classes of words, with the organization of the letters defining words. And since you're inventing the language, I guess you can set the rules. ☺

If you really mean "morphemes" or "phonemes", then yes, of course you could design your language this way. One straightforward example is the Morse code, composed of three symbols - dot, dash, pause. If you e.g. represent the dot and dash vocally as two different sounds (different duration, different pitch or whatever, or even the traditional dit and dah), you can plausibly claim your language has only two phonemes (Though I would call it only a code of the main language it is used to encode, since it lacks its own grammar and vocabulary. But then, there are Q-codes, so nothing is absolute.).

If you want to use three syllables, it would be even easier - after all, Solresol manages with just seven, and Marr's theory claims all (European?) the languages originate from just four syllables (but that is very much a pseudoscience, most famously debunked by no one else than J. Stalin)

If this is a language, then it should be possible to express more than just three words. You would need to be able to say, for instance, "the dog bites the cat." Imagine that we derive dog from sazar, bite from kitiv, and cat from lomon by reversing the consonants. Using only the grammatical information from your post and making some assumptions about syntax, we could form this sentence as razas vitikomivik nomol.

Is this possible? If so, then this is a working language. However, once we define razas to mean "dog," then we already have more than three words. Razas isn't an inflected or declined form of sazar; it's a new word formed by derivation. Even meanings related to the original word are a form of derivation (so making razas mean "fire-creature" and using it for all animals still won't help). Once you create new words by derivation, the language has more than three words. In the language you described, every word is originally derived from three roots or base-words.

As you describe it, the language is entirely possible. There's an infinite number of morphemes in the three roots, if you allow for very long words and a lot of repetition. You have 96 syllables, so to use your possible syllables most efficiently, the 96 most common morphemes/words could have a syllable structure like az or iv, your 96² next most common morphemes like azaz or aziv, and your 96³ next most common like azazaz or azivaz, etc. But your vocabulary has to be limited to make sure that your longer morphemes (e.g. aziv) not be confused with combinations of shorter ones (e.g. az + iv) allowed for by your agglutinative morphology when they're not intended to be the same morpheme.