The world's deepest gorge is what?

Where is the World's Deepest Gorge, Canyon, or Valley?

The answer depends upon your definition. According to the Guinness Book of Records 2005:
· "The World's Deepest Canyon" = Vikos Gorge in northern Greece is the world's deepest canyon in proportion to its width, and at one point measures 2950 feet (900 meters) deep and 3600 feet (1100 meters) wide from rim to rim. Its depth is an impressive 82% of its width at that cross-section (depth/width ratio=0.82). "Gorges in many countries have higher depth/width ratio, but none are as deep." My research agrees with Guinness, plus I would like to add some comments:
§ A different area in Vikos Gorge is 5,927 feet (1,780 meters) deep, measured from the top of Papigo tower above Vikos village to the small chapel of "Panagia" at the springs, but that spot is wider and has a less impressive (lower) depth/width ratio than the location quoted above in Guinness Book of Records 2005.
§ Many gorges are deeper than Vikos Gorge but are significantly wider, giving them a smaller depth/width ratio, such as the following canyons: Colca Canyon, Peru (10,500 feet deep); Kings Canyon, California, USA (8,200 feet deep); Hell's Canyon, dividing Oregon and Idaho, USA (7,900 feet deep from Devil Mountain down to the Snake River).
· "The World's Deepest Valley" = Yarlung Zangbo Valley, in Tibet, with an average depth of 16,400 feet (5000 meters) from rim to river, and 17,657 feet at its deepest point. Here, the Yarlung Zangbo river at 8000 feet elevation (2440 meters) separates the peaks of Namche Barwa (25,436 feet elevation) and Jola Peri (23,891 feet), which are 13 miles apart (depth/width ratio = 0.26).
o Although not mentioned in Guinness Book of Records 2005, note that the wider Kali Ghandaki Gorge in midwest Nepal is 21,000 feet deep and 18-miles wide, located between Annapurna (26,503 feet) and Dhaulagiri (26,811 feet). (depth/width = 0.22)
· "The World's Largest Land Gorge" = Grand Canyon of the Colorado River, Arizona, USA, is 277 miles long with an average width of 10 miles and depth of 1 mile.

As the question simply asks for the deepest, there seems no need to go into depth/width ratios. It's like asking for the plane with the greatest wingspan, and going on about aspect ratios. Keep it simple!
So the answer would appear to be
"The World's Deepest Valley" = Yarlung Zangbo Valley, in Tibet, with an average depth of 16,400 feet (5000 meters) from rim to river, and 17,657 feet at its deepest point.

There's a difference between a valley and a gorge. A gorge is expected to have extremely steep sides, while a valley is expected to be much gentler in profile. Other than the location, I'm not sure of the difference between a gorge and a canyon.

Kali Gandaki valley: deepest gorge of the world
Located in Northern Nepal

Actually, the world's deepest "gorge" is located in Dasu Patan, in the Kohistan district of Northern Pakistan. They lie in the Great Himalayas and the gorge is over 6,500 metres deep. It was created by the eroding action of the River Indus which flows through the area. In fact, it is so high, that even if the highest peak of Europe, Africa, North America and Antarctica were to be placed in it, they would still not rise beyond the top of this gorge.

Source: https://en.wikipedia.org/wiki/Great_Himalayas

A canyon is a deep, narrow valley with steep sides. "Canyon" comes from the Spanish word cañon, which means "tube" or "pipe." The term "gorge" is often used to mean "canyon," but a gorge is almost always steeper and narrower than a canyon.

http://nationalgeographic.org/encyclopedia/canyon/

The Vikos Gorge is a gorge in the Pindus Mountains of northern Greece. Vikos is listed as the deepest gorge in the world by the Guinness Book of Records among others.

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

The Kali Gandaki Gorge or Andha Galchi is the gorge of the Kali Gandaki (or Gandaki River) in the Himalayas in Nepal. By some measures the Kali Gandaki gorge is the deepest canyon in the world, being 5,571 m or 18,278 ft lower than Annapurna I which bounds it at one point. Gorge depth is difficult to define due to disagreement over rim height so this claim is disputed.

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