If we put an metal electrode in water with no metal ions present potential should tend to minus infinity? If we consider an equilibrium between metal electrode and its ions in solution we know that equilibrium potential is proportional to activity of ions in solution from Nernst equation. Since this dependence is logarithmic, what we conclude is that when activity of ions tends towards zero, potential tends to minus infinity. If we do place a metal electrode in pure water, would its potential decrease continously if we disregard any kind of other processes on it like corrosion which create their own potential?
Yes. The potential is in principle infinite if the water is pure. But in reality, when a metallic plate is dipped into pure water, you cannot prevent one or two atoms from being oxidized by air and pass into solution. On the other hand, the same thing can be said about $\ce{O2}$ producing one or two $\ce{OH-}$ ions. As a consequence the potential goes quickly from infinite to a finite value.