That question was posed on the latest internet darling website, Quora.
Some of the answers:
Tongariro Crossing – New Zealand
Yosemite Falls – California, USA
Sentiero Azzuro – Cinque Terre, Italy
Table Mountain – Cape Town, South Africa
Many Glacier – Montana, USA
Uluru, Kata Tjuta, and Kings Canyon – Northern Territory, Australia
Corsica – France
Caledon Natural Area – Virginia, USA
Zion Narrows, Utah
Silverback Gorilla hike in Rwanda
Half-Dome, California
Great Wall of China – Simatai to Jinshanling
Havasupai, Arizona
Dubrovnik, Croatia up Mt Srd to the Napoleonic fort
hike the Macedonian side of Lake Ohrid
Cerro Chato in Costa Rica
Rim to rim, Grand Canyon
Yosemite FallsSuggest
Avalanche Pass, Adirondacks, New York
Table Mountain, Cape Town, South Africa
Corsica, France
Sentiero Azzuro, Cinque Terre, Italy
Columbia River Gorge, Oregon
You can weigh in with your own answers on Quora.
Here’s Table Mountain …
You can climb up, take the cable car down.
besthike.com 
Quora wants me to set up an account, which is too much trouble, so I’ll Comment here.
Most of the responses there were for people’s “favorite” hikes. Come to think of it, that’s an important distinction: I can’t say what the “best” hikes are, since I haven’t done everything in the world. Plus, just as in evaluating movies, “favorite” can be for personal reasons, while “best” should be for more universal.
Thus liberated, here are my top-of-head “favorite” day hikes (“day” also being an important distinction), in no particular order (that would require too much thought), and “hikes” means no climbing required (most of my real favorites are considered climbs):
Buckskin-Paria Loop; S Utah
Half Dome (PR 2:47)
Zion Narrows (my entire list could easily be located in Yosemite and Zion NP’s!)
Huayna Picchu – seen in every photo of Macchu Picchu
Angels Landing or Observation Point – Zion (:46 r/t on the former)
The Subway – Zion (on the border of being a “hike” or not)
Yosemite Falls
Mount Washington via Crawford Path – super historic
Tongoriro Crossing (Circuit is better, but not officially a day trip)
Kaibab – Bright Angel Loop (better stuff in the Grand Canyon, but they’re not considered day hikes)
Via Ferrata’s in Italy – can’t remember their names
Cable Car hikes out of Chamonix – can’t remember names
Hoh River – Sol Duc Hot Springs – Olympics
Chasm Lake (Longs Peak is obviously better, but not sure if it’s a “hike”or not)
Sykes Hot Springs – Big Sur
Thanks, again, Buzz.
My Top 5 favorites:
Harding Icefield, Alaska
Fitz Roy Circuit, Argentina
Eagle Creek, Oregon
Zion Narrows, Utah
Mist Trail/Half Dome, Yosemite
The cliff hike from Bondi Beach to Coogee Beach (Sydney Australia) is high on my list…