On the orchard fonts, trees need headroom to grow, so their rooms need to be multiple z-levels tall. Minimum 2z to not die as sapling, 8z+branches (tallest thus I've seen something like 11z iirc?). You can also, say, place gathering zone on first level of branches and fourth level of branches by building stairs.
A mister would spread mist out more in a multiple z-levels tall room, as mist propagates orthogonally and this allows it to go up and down.
In case of werebeasts, building destroyers can't destroy something that rests on a single wall tile (surrounded by ramps or whatnot), so they greatly benefit from multiz rooms to keep producing stuff in workshops (deconstruct surrounding floors/walls on z-level under). Also, while not technically what you think of, if your hospital individual room entrances are through a hatch you can just lock the hatch in case the dwarf got bitten, thus giving a benefit to 2z hospitals. Or 3z, if you also include mister, well or the like.
Plus, from room value perspective entering the room through a hatch in the room results in more valuable rooms than entering through the door (and also allows you to stack dwarves bedroom/dining/etc. rooms vertically, although typically only nobles get anything other than bedroom).
When irrigating, you need to bucket water from at least 1z above. When bucketing magma sea into obsidian, it needs to be 2z above, since otherwise the 1/7 boils away. When obsidian farming, you also need at least 2z.
When doing a cave-in trap, you'd typically need at least 3z - bogie, support above bogie, and floor above support. But this is quite stretching the meaning of room.