Contrary to what many people might think, the Merced River is not the biggest river in Yosemite, and Yosemite Valley is not the deepest canyon in the national park. The honor of the deepest canyon and biggest river belongs to the Tuolumne. It is not a surprise that the Grand Canyon of the Tuolumne River is not well known. It is a roadless, and sometimes trackless wilderness that is seldom seen by park visitors. A drive to Hetch Hetchy reservoir gives only a hint of the huge canyon upstream, which in some places is 2,000 deeper than Yosemite Valley. The canyon of the Tuolumne also hosted the longest glaciers that ever existed in the Sierra Nevada.
I've only been able to visit the fringes of the deepest parts of the Tuolumne, but here, courtesy of Leor Pantilat, via the Yosemite Blog, is a beautiful video of a run down the canyon.