How many Pomeg-type berries (i.e., the ones that lower a state but increase happiness) would I have to feed a pokemon to get it to 250+ happiness?
The formula I think is the following:
From 0 to 99 friendship, each berry gives +10.
From 100 to 199, each berry gives +5.
From 200 to 255, each berry gives +2.
If a pokemon is caught in the wild or traded, it starts with 70 friendship. If it is hatched from an egg, it starts with 120 friendship. So that means, you need 51 berries if you start with a wild/traded pokemon, and you need 44 berries if you start with a hatched pokemon. (That's to max it to 255. If you want just 250, then you can forgo the last 3 berries.)
Getting the berries in X/Y is not very difficult once you have the first one since the berry fields let you get a lot of them very quickly.
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But if you are looking for the quickest way to max friendship, there are two methods and one extra mechanic you should take advantage of
Leveling-Up:
From 0-99 friendship, each level gives +5.
From 100-199, each level gives +3.
From 200-255, each level gives +2.
Super Training:
Soothing Bag always gives +20 to friendship.
Extra Boost mechanic:
If a pokemon ever has its friendship increased in the same location it was met, then it gains an additional +1 friendship to each boost. (So, for instance, level-ups would give +6/4/3 to friendship.)
So the absolutely quickest way to max friendship on a hatched pokemon is to do the following:
(1) Hatch the pokemon in Lumiose City. (Friendship = 120)
(2) Level up the pokemon in restaurant battles to level 40. (This gives it 39 level-ups. The first 20 give +4 and the last 19 give +3.)
Since you only need level 40 to get max friendship (as long as you hatched it in Lumiose City), then you can feel free to first EV train the pokemon outside of restaurant battles (ideally in horde battles), and then come back to Lumiose City for the additional levels.
If you do not level up the pokemon with Lumiose City at all (or do not hatch it in Lumiose City), then you need 54 level-ups. Typically, if you follow my guide earlier in this thread about power leveling, then you can get about 55-60 level-ups in two rounds of restaurant battles.
That's certainly quicker if you are maxing friendship of many pokemon since you would otherwise need 44 berries for each one (or just 31 berries if you feed them in the same place you hatched the egg). Soothing Bags are not common either, so I don't even consider those. But if you do happen to have them, only use them once the pokemon has 200+ friendship to minimize the overall time and resources.
However, if you are trying to max the friendship of a pokemon traded from Pokebank (which is likely already at level 100 or at least high enough where leveling is not very fast), then feeding it berries is the quickest way. In that case, you also cannot get the extra boost, and so it will take the full 51 berries. I had to do this with my Pinsir and Kangaskhan on the cartridge to maximize the power of Return, and it was not fun amassing all of those berries. (In my case, I did manage to have 4 Soothing Bags saved up, but it's still a lot of berries.)