I think the rarest is probably:
1: The dragon egg: You can only get one per world, it's a challenge to get to the end, and even then, you have to beat the dragon.
2. (I know you can't technically place this, but I think it still should count) Silverfish spawners/silverfish blocks: You have to use eye of enders to find a stronghold, which contains a single silverfish spawner. There are only three strongholds per world, so there are only three silverfish spawners per world. Silverfish blocks only spawn in strongholds, and not naturally, so unless you set up a silverfish farm from the spawners, it is possible to make silverfish extinct.
3: Mossy/cracked stone bricks: Again, there are only three strongholds per world. It would definitely be possible to rid the world of Mossy/cracked stone bricks, it would just take a while to mine them all.
4: End stone: There is one end island per world. Even though that world has probably at least 100,000 blocks or more of endstone, that is still all there is.
5: Circle stone brick: These you can only find in jungle temples. And, as far as I remember, there are only three per jungle temple. Temples are quite rare, which makes this block hard to come by. Even with those statistics, this is still ranked below things like endstone, because there truly is limited numbers of those. If you want more circle stone brick, you just have to venture farther out in your 60,000,000 x 60,000,000 world to find more temples. Even with how rare temples are, there is still probably more circle stone bricks than there is endstone.
6: emerald ore: It only spawns in extreme hills biomes, spawns in veins of one, is slightly rarer than diamond, and is spread out between more layers of diamond, making it harder to find. Need I say more?
Things that actually aren't rare even though they may seem rare:
1: Certain mob drops: All mob drops are renewable, aside from the dragon egg. Ghast tears, magma cream, blaze powder, ender pearls, etc. may seem insanely hard to get, but they are renewable. The only mob drop that isn't renewable is the nether star being added next update because it requires soul sand to spawn the Wither, which isn't renewable.
2: Mycelium: It might feel impossible to find a mushroom biome, but once you've found one you essentially have got nearly infinite mycelium. Silk touch a single piece of it while you are visiting the mushroom biome, and take it back to you base. Once there, find a place grass can't spread to, and place dirt. Place the mycelium down, and place some dirt near it. Once it has spread, silk touch it. You can now, if you feel like it, turn every piece of dirt you've got into mycelium, if you feel like it. It isn't renewable though, because dirt isn't renewable.
That's all I've thought of for now. If I think of more, I'll add it to this post.
Dragon Egg: Duh, there's only one. Nether Stars: You have to kill the Wither (easier than it sounds). Mycelium: You have to find a mushroom biome and have Silk Touch, but once you have some it's easy to get. Dirt: Good, you're still paying attention. Chiseled Stone Brick: You have to find a Jungle Temple, which there are only 3 of. Cheese: Because it doesn't exist.
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Did something happen to you in your childhood to give you this unreasonable fear of rutabaga?
Everybody says about cocoa beans, but aren't they just grow on jungle trees in jungle biome?
As for dragon egg. Yes it's only one, but you know exactly where it is and how to obtain it, so it's not actually 'rare'. It's just a matter of quite a certain ammount of time to get it.
As for things i saw just once:
1. Zombie pigman in overworld (spawned next to nether portal)
2. Record (i don't actually know how to get'em on purpose. It's just one creeper dropped it randomly)
3. Stronghold with all it's stuff (found one by accident, but using ender eyes it's not so hard to locate)
As for things i've never seen (not including end stuff wich i never visited)
Charged creepers, pink sheep, mooshrooms (Rarest mobs, don't say enderdragon, because everyone knows where to find him), pig turned into zombie pigman.
In terms of blocks, e.d. egg (just realized the enderdragon abbreviates to ed. I'll call him that from now on.)
Chiseled stone brick (get the name right, guys)
Natural non-normal sandstone (I think that's only desert temples...)
In ores, I would probably say emerald ore. They only spawn in extreme hill biomes/ jungle temple. The rarest BLOCKs though, would probably be the indestructable ones, since they are incollectable. In armor, probably chain armor. In potions, the bottle o' enchanting, which are rarely traded by villagers. In enchanted equipment, I would say enchanted hoes.(If that even exists.) In mobs, not including bosses, probably blazes.
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This forum needs to be 20% cooler. :/
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Lighting creepers; Spider Jockey with enchanted bow; Ender dragon eggs; Double dungeon/ Triple/ Quad; Music discs and golden apples found in dungeon chests; Ghast tear, if you don't have a ghast farm; Poisonous Potato; Enchanted Looting and Fire Aspect golden sword dropped by Zombie Pigman.
The Meaning of Life, the Universe, and Everything.
Dragon Egg: 1 per End World (this is deletable which regenerates it)
End Stone: Fininte amount of it (other than regenerating the End)
Mossy and Cracked Stone Bricks (finite amounts of this without primary world resets)
Chiseled Stone Brick: 3 per jungle temple and uncraftable.
Nether Stars and beacons are quite hard to get as well, though infinite
Sponges! :3 Nah, the only thing in my SP world I do not have over a stack of (besides Dragon Eggs) is Mycelium. I found tons of mushroom biomes but never got the stuff cuz im lazy so technically for me it isn't rare. Here's a list: 1.) Dragon Eggs 2.) Pink Sheep wool from sheep that spawned as pink. 3.) Obviously Cobblestone...