I have here a medley of Phantom of the Opera tunes arranged for the piano and played by Newell Oler. It runs about 23 minutes. I’m sure there are much longer piano-only tunes, I just don’t know what they are.
So, what’s the longest work for solo piano, meant to be played in a single sitting (i.e. breaks between movements or something would be ok, but an ACTUAL break, like intermission during a performance, would not be)?
I’m not talking some John Cage-ian, avant gard, performance art stuff, either, I mean mainstream classical (or not, but mainstream) music.
The one that comes to mind is Bach’s Goldberg Variations. When I played them in concert, they ran 56 minutes and a handful of seconds. They are meant to be played in a single sitting, yes.
They were also meant to put Goldberg to sleep. Go figure 
Tchaikovsky’s Piano Concerto #1 is of nontrivial length. Dunno of any recordings that stretch more than the 56 minutes that Elenfair cites for the Bach, but there’s a lot of it and any short chunk of it is a nontrivial effort to play.
La Monte Young’s Well Tuned Piano, when played by the composer, lasts over six hours. No break.
Kaikhosru Sorabji’s Opus Clavicembalisticum is intended to be somewhat shorter, around the 3-4 hour mark (with two optional intervals), but performances tend to take 6. This was the reason Sorabji refused permission for performances to take place for most of his life. FWIW, it was played last year in London and last week in New York by Jonathan Powell, who certainly failed to keep in the correct time frame.
Satie’s Vexations is two lines of music, to be repeated 840 times - a twenty-four hour effort, which was performed this January in the Barbican, London, albeit changing performer every fifteen minutes or so.
Damn, I realised you didn’t want ‘avant garde’. So only the Sorabji stands 
Oh, and if you want concertos: Busoni’s is something like 90 minutes, with the longest ‘repertoire’ concerto being Brahms 2, at around 50 minutes. Lightweights 
Another repertoire piece - Messiaen’s Turangalila Symphony, a good 1 hour 40 mins, has a prominent piano part throughout, althought this is stretching the envelope a little