Berezovsky plays Islamey, by Balakirev Composed in 1870 or so, for years this was regarded as the hardest piece of piano music written. This youtube clip apparently has about the fewest wrong notes of any. :)
Ilya Petrov plays Feux Follets, by Liszt Of all Liszt's music, his Transendental Studies are the most notoriously impossible. "Feux Follets" is my favourite.
Marc André Hamelin plays Alkan's Symphonie for solo piano, Finale Alkan is really obscure, so I was surprised to find any clips at all of this online. How anyone plays this stuff is beyond me.
Maria Mazo plays the fugue from Hammerklavier Sonata op.106, by Beethoven The last movement of the Hammerklavier sonata is Beethoven's most technically difficult piece for piano, and was regarded as unplayable at the time. And he was so deaf when writing, there's a story of him holding a pencil between his teeth with the tip against the piano, in the hope of the hearing something through the vibrations.
Art Tatum, "Tiger Rag" Of all the jazz piano masters, Art Tatum had the most jaw-dropplingly unbelievable technique. There's only audio in this clip.. but, yes, there is just one person playing the piano.
Jorge Bolet plays Chopin/Godowsky Study No.12 For Godowsky, apparently Chopin's studies were too easy, so he rewrote them. According to Wikipedia, the 53 studies "are so taxing, even for virtuosi, that only three pianists have ventured to record the entire set"
Ivo Pogorelich plays Scarbo, by Ravel The last movement of "Gaspard de la Nuit". Ravel's explicit intention was write something harder to play than Balakirev's "Islamey". And he succeeded.
Martha Argerich plays Rachmaninoff's Piano Concerto No. 3 Made famous by the movie "Shine" as the piece of music Australian pianist David Helfgott suffered nervous breakdown trying to learn to play.
Oscar Peterson, "Mirage" The only other jazz great who's technique was up there with Tatum's.
Glenn Gould plays Goldberg Variations by Bach (Aria & Some Canons) This is the odd man out. It's not that technically difficult, but it gets a mention because of Glenn Gould's freakish genius. He was a total nutjob.. singing while playing, insisting on sitting at the exact same chair & piano when performing, and surviving most of his life on milk, scrambled eggs and arrowroot biscuits. He also seems to attract the most retarded comments on youtube. But he's usually seen as the best player of contrapunctal music of the 20th century.