Your description still leaves more than 1 possiblility...
You probably have been to http://absastro.tripod.com/data.htm

If this is one of the Begonias supermarkets regularly sell, there is every likelihood that someone knows what it is, if only you had a picture.

here are some other sources of pictures:
http://pageperso.aol.fr/novaeee/photo%20main.htm
http://www.begonia.rochefort.fr/IndexBot.htm
http://www.begonia.rochefort.fr/IndexCV.htm
http://www.hodnik.com/catalog/default.php?cPath=25
http://merseywebdesign.co.uk/clients/begonias/begonias.htm
http://www.golatofski.de/Pflanzenreich/gattung/b/begonia.htm
http://glasshouseworks.com/trop-b.html
http://glasshouseworks.com/trop-b2.html
http://logees.com/products.asp?dept=200
http://seeds.thompson-morgan.com/us/en/list/full-index/b/5
http://seeds.thompson-morgan.com/us/en/list/full-index/b/6

I just bought a 'Reiger Begonia' in my supermarket that sounds exactly like what you've described. It had the tag so that is how I know the name. You can look up on Google Image search for more pictures. Reiger Begonia's don't like water on the foliage. I've been watering mine once a week and it seems to be doing fine. Have it in a SW facing window.

Here is a link that might be useful: Some pictures on the web...

That's it!!! Rieger Begonia! I need one in every color! Does anyone know how to do this without buying them? LOL Seeds? How long for a seed to become the little 4 inch potted beauty? How long do they last?? I think it would be a half hardy annual...can I keep it going as a houseplant? or is there a point of "no return"? The thrill of noticing a new plant!! Thanks for your help! LC Grace

GardeningWithLCGrace - I think you can take cuttings of the plants and root them to get more plants. In my supermarket the plants were only $3.50 which was a bargain. Were yours much more than this? So dealing with the seeds seems like it wouldn't be worth it...

BTW You are right it is spelled Rieger not Reiger (I rechecked the tag) but on the internet I've seen both spellings...

Good luck - they are fun little plants. My mother in law put hers out on her shaded deck all summer and it was so huge and beautiful. That is why I bought the plant!

Here is a link I found that may be of help. I am sure there are better links out there. This one found with the incorrect spelling...

Another one:
http://www.denverplants.com/flower/html/begon_rie.htm

Here is a link that might be useful: Care and propagation Link

The price was about the same as yours from what I remember...I want to go back and purchase 1 of each color! lol I'd like to figure out how to propagate also. So many things to learn! I'll be spending the afternoon checking out the links that you've given...thanks so much! Happy Gardening! (On the windowsill!) LC Grace

I just bought two for $1.50 each! Got nice orange one and a nice red one!

Riegar Begonias can be kept to over winter also.I have a few and all I do is cut them down semi dry them out and they come back the following year.I keep them in the house or my greenhouse.***Suzanne