Some interesting challenges are (use other world or inv editor to get items):
1: Play through the entire game upside-down using a gravity globe (unless using real gravitation potions). You fail if you fall into the sky.
2: Play through the entire game with the neptune's shell and the fishbowl equipped, so you always drown out of water (when you use a gills potion, you can switch your helmet to something more useful for the duration of the buff)
3: Play through the entire game underground, from when you first spawn. Dig down immediately, or walk into a cave if you see one, and never come outside - or close to the surface - again. Don't bother chopping trees either.
These are especially fun on hardcore mode ( ͡° ͜ʖ ͡° )