"Il a même plus de pommes."
There seem to be several instances of the same question: why "de" instead of "des" (or "du"). Here's what I think is going on.
First, the sound. It sounds like "de pomme" rather than "du pomme". Since pomme is fem., that suggests silent s's. Hence, it should be "des pommes".
However, there are rules for when the "des" becomes a "de". The most common one is when there's an adjective in-between "des bons pommes" would become "de bons pommes". That isn't the case here. BUT: as this page indicates,
when the noun is unspecific, it's "de" and not "des". See the examples in the two columns.