I saw several mentions in this suggestions forum for triple bunk beds. I thought Ana's Classic Bunk Beds plan was a great starting point, which gave me a couple ideas.
I'm still trying to get the hang of Sketchup, so I just did a little copy and paste to mock up some examples.
If you just add a layer to the current design with the same headroom and crop the legs, the resulting frame is close to 8 feet tall:
That would require nearly 10' ceilings to have the same headroom on the top bunk. If you reduce an inch between each ladder rung, that cuts headroom by 3 inches, but means you only need a 9' ceiling to have the same clearance on the top bunk.
Bear in mind, that leaves the distance above the top of the mattress at about 25" (depending on mattress thickness), which would be a head banger for any kid taller than about 4'10".
If you do a stagger design, you can fit it under common ceiling heights and still have headroom by making the clearance shorter over half of the bed, presumably the end for their feet:
In this design, I accounted for the front and back panels to separate into 2 pieces, which keeps all the pieces smaller than 5' x 6'. It would also require 2 more intermediate cross panels similar to the end panels.
In either design, I think I would change the verticals to all 1x4s for added strength.
I also have an idea for an L shape staggered design. Not sure when I'll find time to play with Sketchup, but if I get some encouragement maybe I'll try to draw up one of these.