Does there exist a theorem whose known proofs encapsulate all possible proofs of that theorem? I was just reading a book on number theory which gave ten different proofs of the infinitude of the primes. This caused me to wonder whether or not it would be possible in principle to find every proof of the infinitude of primes which in turn caused me to wonder if there exists a theorem whose known proofs encapsulate all possible proofs of that theorem. If this question is complete nonsense please let me know.

If there is one proof, then there are infinitely many, since you can always add trivial steps. You see this really clear in formal proofs: once I have a statement $P$ I can always derive $P \lor Q$ for any of an infinite number of statements $Q$, before getting to the theorem.
Of course, one can complain that adding nonsense steps doesn't really change the 'core' of the proof, but making that kind of thing hard, i.e. trying to determine when one proof the really is 'different' from another becomes mostly a subjective exercise.
