Can you have a permanent become all permanent types at the same time?
Yes

Play Gideon Blackblade. He will be both a creature and a planeswalker on your turn.

Play Mycosynth Lattice to turn it (and everything) into an artifact.

Play Enchanted Evening to turn it (and everything) into an enchantment.

Play Xenograft, naming saproling, to give all your creatures the saproling creature type.

Play Life and Limb to turn the Gideon Saproling into a land (forest) as well.

I'm sure there are other ways to do it, but this might involve some of the least complicated rules.

It's worth mentioning that there are other types that can apply to permanents:
Basic, Snow, Tribal, and a few other silver-bordered ones.



If you add a March of the Machines, everything is a Artifact Enchantment Creature, and your permanents are Artifact Enchantment Creature Lands - Forest Saproling. Be careful, as this will probably destroy your lands (well, your land cards in play, since they have CMC 0 and will be 0/0 creatures. I believe Forests will be OK, remaining at 1/1 creatures.)
Any planeswalker you play from that point on will be a Legendary Artifact Enchantment Creature Planeswalker Land.