Is the full stringy potenetial (for which it is claimed to be 10^500 vacua) written down explicitly somewhere? Any references? Thanks, Dave
String theory isn't just a simple quantum field theory with finitely many fields so it is not true that the "string theory potential" is a function of a fixed number of scalar fields (or other fields). Instead, at various points of the configuration space, the number of light (and even massless) scalar fields is changing, at various points of the transitions etc. The configuration space is very complicated.
So no, there's surely no compact formula for the "string theory's potential". To write such an object correctly and globally, one must really define all of string theory and take it into account.
With regards to the follow-up comment: there is no example that covers the most important features, because the only framework where you might understand how to think about a potential with multiple minima is QFT. But there is no evidence that supports this approach in String Theory, and lots of evidence that suggests it is just wrong. Instead, the only situations in String Theory where we truly know what we're doing all suggest that different spacetime asymptotics correspond to different theories; not states in a theory that might be explored by something like tunneling.