This beautifully veined plant grows in the shade under the tree canopy in the rainforest, sometimes in large groups. It was not in bloom the time I was there.
It has winding stems.

Cissus discolor? (Rex Begonia vine)

Forgot the link to the database entry.

Rex Begonia Vine (Cissus discolor)
Apparently mine was tagged with a synonym. Thought I'd taken a pic of the blooms, but I don't see it uploaded. They're not showy, even by my appreciation for tiny flowers.

Thank you Tiffany! I see the similarity so maybe mine is in the same family..

I was figuring my store-bought plant was some fancy cultivar, but wow, there's a ton of Cissus species in the plant database, though very few have any pics. Yours is different enough that I agree, probably a different species than the plant I bought.

This is probably more likely.
Amazon Jungle Vine (Vitis amazonica)

We're still not there Tiffany, but thanks anyway for trying to help!

YW, 'twas educational for me too - in a pump-up-the-wish-list kind of way! If you can find a source of pics of more species of Cissus, you may find the right one if nobody who happens to know pops by this discussion. If you do find tons of pics somewhere, I'd love to see them too! So few really pretty vines grow in the shade, though knowing about them doesn't make them any more available.

I doubt it is a Cissus sp., I have checked most of the species found in Brazil and there was nothing like this.

Iresine herbstii does look close .. the leaves look a little dentate in the first link ..

http://luirig.altervista.org/f...

http://www.ceapdesign.com.br/f...

http://www.exoticgarden.com/bl...

http://www.dkimages.com/discov...

The leaf margin on your plant isn't entire Myriam, but maybe it's in Amaranthaceae.

http://www.plantillustrations....

Amaranthaceae is very variable, I've looked at all the possibilities for Brazil and found nothing which is supposed to be there so I will post this to rule it out. The base shape of the leaf is different from most in the family.

Thanks so much Janet!

I thought this was a vine plant?

I wrote vine-like plant Tiffany, because it has these winding stems, like it shows in one of the pics I've posted, I haven't seen it actually growing up a tree or any support but itself, and I guess it would do that if it had the opportunity..

I've looked into this a couple more times and can only tell you, it's definitely not the Cissus (any Cissus) I suggested. Your pictured plant has opposite foliage, which negates Cissus. Wish I had something more positive from the effort, but found nada. So just recording this info here for further consideration for those who might have an idea.

Thank you Tiffany for your efforts!

Bump!
I've taken a small rooted piece from this plant back from my last trip in the hope it is going to flower and to give us a clue as to what family it could belong.
In the three months I'm back it grew quite well, now much slower, but still looking good, I think it is investing in root development..
It doesn't let me loose what it could be, it has something mysterious about it, maybe whitchy..

Just jumping in with a guess:
Start with Pilea anisophyllea and/or Pilea pumila and work from there.

Thanks for the suggestion Greene!

Mystery solved!!!
Thanks to Wendy Cutler (UBC Botanical Garden F.) who asked Alex Popovkin on flickr,
who has a magnificent photo collection of the flora in Bahia, more Northern than the Rio state,
https://www.flickr.com/photos/...
who on his turn asked his colleagues from Rio, and got the response that it is the juvenile form of Mikania hemisphaerica.

http://plantillustrations.org/...
The same botanical drawing in the Flora do Brasil:
http://floradobrasil.jbrj.gov....

Wow Myriam, what a great network the internet is! Seek and you shall find.

Interesting the "juvenile" form, which to us suggests a 'young' form.

https://sites.google.com/site/...

My mind springs to another meaning which is a "junior synonym". Under Zoology:

https://en.wikipedia.org/wiki/...)

The name is accepted..

http://www.theplantlist.org/tp...

http://compositae.landcarerese...

There is some doubt now again Janet
The photos of the plant in question on the flora SBS, same link you gave me don't correspond with my mystery plant.

I'd better link to my thread in the UBC:
http://forums.botanicalgarden....

I see silver surfer has found the same plant on Flora SBS which doesn't match at all, not only the silver veins but also the venation differs apart from the shape, but shape may differ. You can enlarge the photos to get a better look.

On the page is:

Bibliografia

RITTER, M. R. MIOTTO, S. T. S. Taxonomia de Mikania Willd. (Asteraceae) no Rio Grande do Sul. Hoehnea 32(3): 309-359, 168 fig., 2005. Disponível em: <http://www.ibot.sp.gov.br/HOEHNEA/volume32/abstract111.htm>.

Ritter, M.R.; Liro, R.M.; Roque, N.; Nakajima, J.; Souza-Buturi, F.O. Mikania in Lista de Espécies da Flora do Brasil. Jardim Botânico do Rio de Janeiro. Disponível em: <http://reflora.jbrj.gov.br/jabot/floradobrasil/FB5453>. Acesso em: 26 Set. 2014.

I had to copy and google the first link, this is what it brings you to: Mikania vitifolia

https://sites.google.com/site/...

Also: Mikania hirsutissima with the same browser address ..

https://sites.google.com/site/...

https://www.google.co.uk/?gws_...

The second link on Mikania hemisphaerica only worked in the browser address, otherwise it gave the same page the link was on. It takes you to the entry of Flora do Brasil.

http://reflora.jbrj.gov.br/jab...

I tried the links in the first one but got errors or oops. Maybe the addresses for pdf's would work if copied and pasted into the browser, some only work that way.

"Disponível em" translates to "Available in". There's several links in the second link for Mikania hirsutissima which might help.

Translation of text can be done by highlighting the text, right click and choose "translate with bing".

The google search also brought up a page of links ..

https://sites.google.com/site/...

http://www.bing.com/translator...

The last in the google search has lots of links which might work if done in the browser.

http://web500.com.br/Garden/?p...