When you took babies off it to give to your friends, was there a sort of bulb or rhizome on the bottom of the plant between the stem and the roots? Does it go dormant in the winter? Disappear below ground or just stop growing?
I agree with Kay, it looks a lot like a Caladium, and they can make a clump like that if left in one place for a few years, but the leaves look thicker and shiny, and to be honest, I've never seen a plain green Caladium.
Might be an Aroid, some sort of small Philodendron, or an EE. Maybe post your pic on the Aroids forum?