Easy:
Automate the production of titanium from rutile ore, limestone, fuel and water.
Kill a pigman and don't die from the other ones.
Feed a resonance chamber an Aeternalis Fuel Block.
Incinerate a forest with the Sky Ember ability.

Medium:
Cast a curse on yourself (Why would you do this? Figure that one out yourself.)
Fight the Wither without wearing armor. Don't use Avaritia stuff for this one, that kinda defeats the purpose.
Gather infinite ores with ChromatiCrafts mineral enrichers. Make this system self-sustaining.

Hard:

Fully automate Thaumcraft ore multiplication, without using Applied Energistics or Extra Utilities, and without inputting resources that are not ores.
Fully automate the creation of arcane prisms.

Lunatic:
Do any of the Botania Lunatic Challenges. Feel free to cheat a little and use other mods, but don't use Applied Energistics or Extra Utilities.
Automate the production of Aeon Steel from raw resources, as far as possible.
Breed a Celestial Bee.