If we assume there is no limit to the amount of mathematics we create/discover, is it possible that there are infinitely many proofs about any true mathematical statement? For example, we have Wiles' proof of Fermat's Last Theorem, but due to the deeply interconnected nature of mathematics surely there are many other proofs of it out there waiting to be discovered. Another example is the plethora of proofs regarding the Pythagorean Theorem which seems to be understood better the more we generalize it. In other words, is mathematics an infinitely large web that is connected at infinitely many points? (this is probably an inadequate analogy for capturing mathematics but it's the only one I could think of)
Suppose I have a proof of $\varphi\implies\psi$. I will now construct infinitely many proofs of $\phi\implies\psi$, where $\varphi$ and $\psi$ are propositions. The first one goes like this:
Here's the second one.
I can do this as many times as I like, and get a different proof of $\psi$ that way.
In case you're unfamiliar, $"\land"$ means "and."
The thing is that none of these proosf are functionally different. It's just a cheap way of creating a "technically" different proof. If this is what you mean by "are there infinitely many proofs of any statement?" then yes.
What you might mean, though, is whether there are infinitely many proofs that "use different ideas" or use some different kind of insight. This isn't really a precise definition, but it's all I've got to work with. The question isn't really answerable until you define what you mean by "different proofs."