What items can be farmed fully automatically?
Aside from the things you mentioned:
Fully Automatic (no need for player's presence):
As long as you keep the chunk loaded, you can go on an adventure while your farm is producing
- Anything that uses a smelter (cooked food, experience generators, etc.)
- Honey Bottles, Honeycomb
Fully Automatic, as long as player is within 128 blocks of farm
- Melons & Pumpkins can be farmed using BUDs.
- Mushrooms are easily farmed, since they spread by themselves.
- Eggs can be farmed
- Wheat can be fully automatic using a farmer villager
- Bread with farmer villagers
- Cactus
- Coal
- Iron ingots
- Gold nuggets
- Steak, Raw Beef, Leather (if you are lucky enough to find a mushroom island)
- Cobblestone with Wither or Ghast
- Cooked Chicken, Raw Chicken, Feathers
- Redstone, Gunpowder, Glowstone Dust, Sugar, Wooden Sticks with a witch farm
- Slime Balls
- Sand & Gravel Duplicators and Flint (actually a bug, but still possible)
- Carrots and Raw Potatoes with farmer villagers
- String, Spider Eyes
- Bones, arrows, bows
- Rotten flesh
- Seeds and any kind of 1 block flower
- Ender pearls
- Any kind of potion
- Prismarine Shard
- Magma cream
- Ghast tears
- Kelp
- Bamboo
- Sugar Cane
- Wool (dispenser equipped with shears)
- Records
AFK farms:
You may need to jam the click button, and you wont be able to play while harvesting
- Snow (at some time pre 1.15, this was fully automatic. No longer.)
- Any experience farm
- Wooden logs (any type)
- Tree saplings (any type)
- Fish, Name tags, Nautilus shells, saddles, enchatment books
- Blaze Rods
- Any kind of 2 block tall flower
- Animal breeding farm
- Dragons Breath
- Cocoa beans
- Nether Wart
- Ice with a BUD to detect the freezing water (not sure if withers can destroy ice)
- Stone (needs silktouch pickaxe)
- Any Type of Wooden Planks just craft the output of the log farm
- Glass and Panes (smelt and craft the output of the sand duplicator)
- Flint and Steel (craft the outputs of the Iron farm and the gravel duplicator)
- Mushroom Blocks
- Grass and Mycelium (although it is not technically infinite, it is practically infinite)
- Emeralds trading the output of another farm if you get a perfect villager
- Full diamond toolkit and armor trading emeralds with a blacksmith villager
- Mob heads except wither skeleton (with a charged creeper farm)
After a little more than a hour looking up things and figuring out stuff, I think I have a list of all items you can make farms for. (Only the automatic farms). Here is the list and hope this helps.
Planted Crops
- Cacti - Cacti farm.
- Sugar Cane - Sugar Cane farm.
- Mushrooms - Mushroom farm.
Mob Grinders
- Iron - Iron golem farm.
- Roses - Iron Golem farm.
- Eggs - Chicken farm.
- Bones - Skeleton farm.
- String - Spider farm.
- Rotten Flesh - Zombie farm.
- Blaze Rod - Blaze farm.
- Ghast Tears - Ghast farm.
- Sticks - Witch farm.
- Glass Bottles - Witch farm.
- Glowstone Dust - Witch farm.
- Redstone Dust - Witch farm.
- Sugar - Witch farm.
- Assorted Potions - Witch farm.
- Gunpowder - Creeper/Witch farm.
- Ender Pearl - Enderman farm.
- Magma Cream - Magma Cube farm.
- Gold Nugget - Zombie Pigman farm.
- Gold Sword - Zombie Pigman farm.
- Slime Ball - Slime farm.
The lists above are excellent!
But, this question was asked before Minecraft 1.8 so a few things have changed.
Now you can use Villager Farmers to automatically harvest and replant Carrots, and Potatoes, and Wheat. You can also automatically collect the Bread that Villager Farmers make with the wheat.
Another source of automated carrots and potatoes are from zombie mob grinders.
You can fully automate Pumpkins and Sugar Cane, but not melons. Any automation will break them into Melon Slices. So Melon Slices can be fully automated.
Villagers can also be farmed. Because you can have them automatically shipped away from the village and stored somewhere else so that more can breed. I manually do this right now but am building an automated system.
I've built a minecraft world that fully automates most almost everything I can think of. I made a video tour of this world that you see at: