At level 12 From This Question you start to obtain great balls when leveling.
I am currently level 6 and have only ever gotten normal pokeballs from pokestops.
Is there at any point (possibly after 12) where pokestops start giving better balls?
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Yes, you receive Ultra Balls at level 20. Note that once you unlock an item, it becomes obtainable at PokeStops.
Yes! We unlock a Pokemon Go great ball at level 12 and an ultra ball at level 20. But I'm really surprised that everyone forgot Master Ball! I was expecting master ball as well in Pokemon Go!