Fun read! I agree with jeeprs that this sounds a lot like the Cosmological argument. As such, you may enjoy reading another argument similar to the one you crafted. Here's a link to the
SEP page on it.
My two nitpicks would be this: It is logically possible that the universe is neither infinite yet wasn't created. Existence could just be without a creator in the same way that a God would just be without a creator -- In essence, existence itself would be the "greatest cause" you're looking for. The metaphysical conception beyond that just mucks things up.
From this, If we take your argument for a non-infinite universe from the 2nd law of thermo as sound, then we should conclude that this would be the "first time through" (and, if we take the 2nd law as sound [which I'm not arguing we shouldn't, by the way], likely the last time through, too!)
My second nit-pick is in your conception causality -- every effect is thought to have a cause, but that doesn't mean that every cause has its own effect, or that every effect must be greater than or equal to its previous cause. I don't question the existence of causality, but I do question the conception of the Cosmos as deterministic machine. I also think it's hard to determine what "greater than or equal to" means when dealing with something as qualitative as causality. In a sense, the notion of cause is the notion of be-cause, in that it explains why something is -- so cause is the statement which satisfies curiosity, rather than an over-arching efficient cause that creates a machine universe. While I think such a cosmos possible I don't think there's a good basis for this inference either deductively or inductively.
Those are my 2 pennies, anyways.