I'm afraid I'm going to have to frame-challenge this one.
For example, it seems intuitive that a spoken language cannot hold too many words without having a way to write them down (imagine having to memorize 100000 words without the possibility of saving them for later reference).
Perhaps surprisingly, this does not seem to be the case! Writing systems don't really affect spoken language as much as one might expect, which is why some linguists consider them separate from the languages themselves. The Homeric epics, the Vedas, the Avestas, the Oral Torah, and many other enormous works were composed centuries before they were written down; in the intervening time, they were passed down through oral tradition. Human memory is capable of amazing things!
On the other extreme, if the number of words is too small (say a few hundred), i.e. the language is not rich enough, then it would make less sense to invent an alphabet.
As far as we know, a true alphabet was invented only once (by the Greeks, modifying the Phoenician abjad). Writing in general has been invented a decent handful of times, but it's usually associated with a need for record-keeping as opposed to any change in the language itself. Most of the time, when a language gains a writing system, it's because missionaries, traders, or other emissaries from literate cultures came in and introduced the idea—again, not usually associated with any change in the language itself.
So given the above, there should be some threshold (depending on the language) on the number of words which would necessitate an invention of an alphabet.
Perhaps most surprisingly of all, there's an axiom in linguistics (as in, it's almost universally assumed to be true, and nobody's ever found a counterexample to it (*)) that no languages are more complex or more complete than others. Anything that can be expressed in English can be expressed in French, Japanese, Swahili, Sumerian, Nahuatl, or any other. Where language came from in the first place is an open question, but nobody's ever found a "primitive" language in the wild.
How many words a given language has is a complicated question—in part because it's really hard to determine what a "word" actually is! But if there's a measurable level of "how many concepts can this language express?", it seems to be a universal constant.
(*) There's one extremely controversial "exception", the Pirahã language spoken in South America, which supposedly violates everything everyone has ever thought was true about linguistics. But only one linguist has made these claims, he hasn't provided much data or evidence for them, and nobody else has been able to replicate his results. So many linguists just ignore that.