I've been thinking

 I've been thinking.  Models of literary infinity.  

The monkey theory.  Monkeys typing will produce the works of Shakespeare, if only you let them type long enough! 

Hjemslev's / Chomsky's idea of an infinite number of sentences in a language. 

Borges's Library of Babel, of course.  

Queneau's sonnets (14 to the 10th power, cent mille millards).  http://emusicale.free.fr/HISTOIRE_DES_ARTS/hda-litterature/QUENEAU-cent_mille_milliards_de_poemes/_cent_mille_milliards.php

Now, Borges's model is the letter, not the word. Queneau's unit is the line of poetry. The linguists' model is syntactic, admitting grammatically correct utterances. 

If you controlled for length, and had a finite dictionary, then the sentences in a language would not be infinite, in purely mathematical terms, but they would be practically infinite, in the sense that the numbers would be inconceivably large. After all, if a sonnet with 10 versions of each line is already in the billions... 

It is impossible to answer the question: think of the largest number that you can, because you could always raise that number to the power of itself, and keep doing that infinitely. For example, 1 followed by a hundred zeros is a commonly given example, but you could raise that number to the power of 1 followed by a hundred zeros.  

Suppose you have a limited language. 10 verbs, 20 nouns, 20 adjectives, etc... Two tenses. A limit number of pronouns and other connecting words.  Now you have syntactic rules as well. How many sentences are there of fewer than 20 words?  It would be possible to generate the possibilities mathematically. I'm sure this has been done.  

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The other idea is that there are a limited number of plots / archetypes in literature. Literature is finite, because human experience is archetypical, not infinite. You could invent a new plot, but it wouldn't be meaningful or satisfying.