Why do we get saturation current in photoelectric effect

So apparently if you increase the number of photons emitted per second to infinity, the photocurrent will approach a limiting value called the 'saturation current'

I feel like we shouldn't get a saturation current because as one electron is emitted, another will immediately (or close to immediately) fill its place just like in a normal current - thus electrons can pretty much be continuously emitted.

Can anyone explain why we get saturation current? Thanks :D