Berni,
I wonder if this is some sort of Begonia hybrid... Are the leaves kind of roundish like the middle-right leaf appears to be? Or more pointed like the one on top appears to be or multi-pointed like the one upside down on the ground appears to be? No flowers ever? A mutant begonia? Look at this Begonia hernandioides pic, does that match up with yours?

We ought to get CostaCoffee over here. She's the real expert on plants...
Regarding begonias and edibility, turns out there was a 1990 issue of the Begonian which addressed this very issue! ;-)
Laferrière, Joseph E. 1990. On the edibility of begonias. Begonian 57:175
Most begonias are sour to the taste, and some people in some areas eat them. This is safe in small
amounts but potentially toxic in large quantities due to the prevalence of oxalic acid in the tissues.
Joseph, our Begonia expert, also had this to say in his study of the foods of plant origin traditionally consumed by the Mountain Pima of Chihuahua, Mexico, in his paper published in the Journal of Ethnobiology1991, where their food choices were analyzed for proximate and mineral content:
Calia aigre (Begonia gracilis H.B.K. ssp. nervipilosa A. DC., Begoniaceae) is
common in moist, shaded locations, and is occasionally consumed by children.
The stalks are juicy, mucilaginous, and pleasantly acidic. Hierba aigre (Oxalis
decaphylla H.B.K. and O. albicans H.B.K. ssp. albicans, Oxalidaceae) is treated the
same way. Shoots of Begonia and Oxalis are acidic due to the high concentration
of oxalic acid, which is potentially toxic in large quantities (Laferriere 199Oc, 1991a).
Fortunately, the children do not consume enough of these plants to cause
noticeable harm. Shoots of O. divergens Benth. are used by the Mountain Pima
as tea to relieve fatigue (Pennington 1973). Cravioto (1951) gave a figure of
3.7 mg carotene per 100 gO. divergens. Shoots of B. gracilis are used in treating
toothache or gum disease (Pennington 1973).
Sounds like it would be safe to eat some as long as you don't eat too much and assuming it is a Begonia... ;-)
--
Sam