I own a few dozen orchids, most inherited from my grandma. They're surprisingly easy to care for, and I love them very much.
This page is sorted by genus, but I'm probably wrong about some.
These grow as a series of vertical bulbs topped with leaves and beautiful ruffled flowers. I've duplicated mine by cutting between bulbs while repotting.
Apparently, it's Brassanthe Maikai "Mayumi"
. It grows much like the Cattleyas and is similarly easy to propagate. Oddly, the petal color fades over time.
This sympodial orchid generates cute baby plants (with roots!) mid-air; they can be separated and grown on their own.
These have a central crown, flat leaves, and an endless variety of flower patterns. I haven't tried propagating them yet.