Pro tem, though, I'll set aside recursion.
Firstly, I'll discuss the syntactic side. I'll assume a stable lexicon that does not allow for diachronic differences. Limited or not, the number of sentences is very large. (I'm saying sentences instead of utterances, since the number of possible utterances is by definition boundless, if the formal definition is to be used...) There is no corpus that will cover all or even most of the sentences that have been generated, are being generated and will be generated. In fact, the probability of each sentence, including the sentences I'm writing now, appearing in discourse asymptotes to zero.
If there were a limit on the number of sentences, then either a) the number of possible sentences generated tends to infinity or b) there is an arbitrary limit on the number of possible sentences which is so high as to be unattainable. The latter violates Occam's razor since it posits the existence of such an arbitrary limit.
I'd also argue that this can also be explained at the morphological level. The number of possible grammatical words generated is not finite. 'New' words, like 'fictionhood', a word I just generated on the forums on the fly, are not ungrammatical, as I did not violate any rules of derivation and used a valid root. The number of words is very large. An argument similar to that for syntax applies.
What allows for boundlessness is, of course, recursion. Adjuncts, for example, are clearly infinite. Similarly, we can utter an infinite string of 'I think he thinks...' before finally writing out a fullcomplementiser phrase. Recursion appears in morphology as well; think the prefix 're-'. Thus I think it is reasonable to assume that a finite set of grammar rules generates an infinite set of sentences.
I'm sure we all agree that rules in language are based on usage. There are only a finite number of people on earth, and each one of those people has used a finite number of words in his lifetime. Thus, we only have a finite pool of usage to draw our descriptive "rules" from. Yet the possibilities are endless. It is likely that no human being had uttered the words, "Colorless green ideas sleep furiously," before Chomsky did many years ago. And yet, these words have meaning in themselves like no other sentence. Also, Pinker explained that there are limitless possibilities to extend a sentence, and yet still wihin the linguistic definitions of a sentence. Just take any sentence and tack on to the beginning, "Who cares that Pinker mentioned that Guinness noted that Faulkner wrote..."