Suppose I start with a big blob of measure in a boring universe, that is slowly turning into universes like ours. Linearity says that the the rate at which universes like ours appear is proportional to how big the big blob of measure is.
In fact, this is crucial to calling it “measure” rather than just “that number in quantum mechanics.”
So if the rate of universes like our appearing is proportional to the size of the original blob, as we make the size of the original blob infinite, we also make the rate of universes like ours appearing infinite. We cannot have a finite number of universes like ours, but an infinite blob of measure turning into them—we can only have a proportionally smaller infinite amount of universes like ours. This requirement gives us back our old limitations about eventually running into a maximum.