What rare items exist in Minecraft, and where can I find them?

What are the rare materials/drops in Minecraft? I'm talking about things you might never find, except perhaps in chests near mob spawners, or by some convoluted collection process.

I already know about saddles, records and slime balls. What other rarities am I missing?

Solution 1:

Unique Item

• Dragon Egg - only obtainable by killing the Ender Dragon.

Rare Items

• Saddles - Only found in chests in dungeons, Nether Fortresses, Desert and Jungle Temples, and NPC villages. Can also be obtained by trading with Villagers and fishing. It can also be obtained by killing a ravager.
• Horse Armor - Only found in dungeons, Nether Fortresses, Desert/Jungle temples, and NPC villages.
• Music Disc - Only gained by tricking skeleton into shooting a creeper with its arrow. Also, discs "13" and "cat" are found in 8% of dungeon chests.
• Name Tag - Only found in chests in dungeons, Nether Fortresses, Desert and Jungle Temples, and NPC villages. Can also be obtained by trading with Villagers and fishing.
• Melon Seeds - Can be found in minecart with chests in abandoned mine shafts. You can also get them from NPCs.

Rare Mob Drops

• Ender Pearls - A rare drop from a rare mob (Endermen).
• Blaze Rods - Blaze are flying mobs in the nether; hazardous to kill. Can be used to make a few other ingredients via Blaze Powder
• Ghast Tear - Prone to falling into lava when you kill the Ghast.
• Wither Skeleton Skull - A rare drop from a somewhat rare mob (Wither Skeleton).
• Nether Star - An item drop from the Wither Boss
• Mob head(other than Wither Skeleton) - Can only be obtained by blowing up the said mob with a charged creeper (creeper struck by lightning).

Rare Blocks

• Emerald Ore - A new block that only generates in one biome, and generates between three and eight blocks per chunk. Used as a currency for trading with NPC villagers.
• Beacon Block - An item crafted from the Nether Star, 3 Obsidian, and 5 Glass.
• Sponge - Found only in ocean monuments and dropped by Elder Guardians.
• Ice - can only be mined with a silk touch pickaxe, can only be made in a snowy biome.
• Packed Ice - A more compact version of Ice, but doesn't melt in Torch/Glowstone light, only found in the rare Ice Plains Spikes biome, and obtainable only with silk touch enchanted Pickaxes.
• Prismarine - Only found in Ocean Monuments. It can be crafted from prismarine shards dropped by Guardians.
• Sea Lantern - Only found in Ocean Monuments. It can be crafted from prismarine shards and prismarine crystals dropped by Guardians.

Other

• Bedrock/Adminium - Cannot be mined, found at bottom of map or top of the Nether. Must be spawned through the console to place.
• End portal and End Portal Frame - 1,152 end portal blocks per world and 1,536 end portal frame blocks. (As of 1.15)
• Command Block - Only obtainable via console command.
  • Command Cart - Also must be obtained by commands.
• Barrier - Obtained by commands.