The Madeira River is arguably* the longest of the Amazon River's 1,000-plus tributaries. It is also the route I was privileged to travel this summer (July 19-22) with my good friend Dr. Miguel Nenevé. It is highlighted in pink on this map of the Amazon Basin -- a path that coincides with most of the route we took from Porto Velho (near its formation) to Manaus (just upstream of its confluence with the Amazon River.
I spent almost a week in Porto Velho before our voyage and two days in Manaus afterward. Photos and videos from the entire experience can be found on Flickr and YouTube through my Madeira Playlists article.
*Lengths of rivers is always arguable, as the Wikipedia article List of river systems by length makes clear. Just as no river has a single source, it is often difficult to determine exactly how to measure the length. Notice in this article that many of the familiar names (such as Nile, Amazon, and Mississippi) are joined by hyphens to names that most readers will not recognize. Hydrologists can often find a longer river by following channels that do not bear the name of the "main" river.
The Amazon River has a thousand named tributaries, about a dozen that measure a thousand miles or in their own right, and and unknown number of nameless tributaries. The most important of these is the Solimões, which most maps label "Amazon" and which Wikipedia does not even mention in its river-length article.
The word "arguably" above refers to the fact that the Tocantins-Araguaia system is mapped as part of the Amazon Basin, but joins it in the delta, not along the main channel. Does that make it a tributary? Perhaps.