What happens after reaching 99x items of a kind?

What happens after 99? 100. Seriously. I've had 100+ berries, and various potions.

So long as you have room in your bag it should be possible to get more*. I've gone a few dozen Pokestops without getting more Ultra Balls, and I have nowhere near 99 of those; they are just the rarest of the 3 Pokeball types. Max Potions, being the rarest of 4 would, I assume, have a lower drop rate than even Ultra Balls (rates may depend on your level).

*caveat: due to the nature of programming, it is possible that the item count would eventually overflow. But given bag limits, the amount of an item necessary to do so is likely beyond what is possible without otherwise breaking the game.

There is no limit at 99, at least for Ultra Balls. This must be just a co-incidence that you seem to have stopped receiving items when reaching 99.