And should it be unprovable, why should it be widely expected? The truth is that as far as you go along, it will always be possible to give a number which has not yet been listed.

In turn this means that the set of numbers occurring in pi can always be subsumed in a set that includes a number not present in pi.

Examples like this from set theory are both the impetus for statements like, "Some infinities are larger than others."

This same sort of example is also how Goedel's incompleteness theorems work. While the theorem does not state that a consistent theory of algebra is impossible, it does state that it is either impossible form one or impossible to prove that you have one.

The digits of pi are also a kind of halting problem: can you write a general program, such that it is known when the program will encounter a given sequence, and stop? The answer is that you cannot.

Finally, it should be said that true randomness is either non-existent or is unprovable. Instead, when we wish to demonstrate that something is random, what we usually do is see if a given set of numbers occurs significantly more often than any other-- and so randomness is usually defined as a set of outputs in which no individual piece is significantly more likely to be output than any other.

(That this is an incomplete account of randomness can be noted by the fact that one test for cryptographic systems is determining whether its output can pass a randomness test. Successful algorithms do produce such output but we know for a fact that they are not actually random, dependent as they are on specific input.)

Monkeys, it can reasonably be assumed, are not random systems at all. (Given that we have great difficulty intentionally designing systems that meet this standard, and monkeys are presumably not the result of an intentional design that was created to produce randomness, this is a safe assumption imo.)

Taken as a whole, then, it seems obvious that monkeys are not random systems, and even if their non-random output is run forever, this is no guarantee that they can produce a given output even if given infinite time.

Pi well may contain every sequence; my overall point does not change based on Pi's status. I was simply giving an example of a case where we can guess that infinite output may not produce all possible variety. We could well have used repeating decimal expansions like 0.9999...to make the same point.

Last, I'd argue that the impossibility of an infinitely varied sequence is partially guaranteed by quantum mechanics: in most waveform equations (or equivalent forms), a small subset of outcomes is not just vanishingly small but absolutely forbidden. That is, you can know that you will never see some part of the system at a given location in a given configuration at that time.

That such forbidden system states exist implies that not all things that are conceptually possible are actually possible; similarly, that it is conceivable that the infinite output of monkeys might recapitulate Shakespeare is not proof that it is actually possible for them to do so.