Yes. And you get the added bonus of Scott LaFaro’s bass playing.
Wow. That’s about the most beautiful solo piano work I’ve heard in quite a while. Wonderful!
And to all you others: Thanks for a lot of great suggestions. I’ve already ordered Chopin’s Nocturnes and some Debussy from the CD store, and tonight I’m going Youtube trawling for a bunch of the other suggestions.
Ignorance has been fought once again 
Tom McDermott is my favorite New Orleans pianist. He plays in a lot of styles, including jazz, ragtime, Brazilian choro and even some Chopin. My favorite CD of his is Live in Paris, which happens to be all solo piano (which is what you’re looking for). Another interesting CD of his is All the Keys and Then Some which has short pieces in each major and minor key.
Monk Alone is a collection of solo piano pieces played by Thelonius Monk.
It’s kind of the exact opposite of ‘simple, unobtrusive’ but you might find you like some of the ragtime music anyway.
This is my favourite, Black and White Rag, as played by the amazing Winnifred Atwell. Possibly the jauntiest piece of music ever written. If you are a fan of ragtime, it’s very much worth tracking down Joshua Rifkin’s CD of the works of Scott Joplin. Fantastic piano playing throughout.
Also, you mentioned liking Satie’s Gymnopedies. Just on the off chance that this isn’t a piece you know, I give you Satie’sGnossiene No 1, which I think is his most haunting piece and sounds like it might be up your street.
I can’t believe I forgot Schubert! Here’s a lovely pieceplayed by Alfred Brendel, an astonishingly wonderful piano player wringing every last bit of emotion from it. Brendel’s got lots of Schubert CDs. all are worth listening to.
No, I wasn’t.Thank you!
One of the most mindblowing piano pieces I’ve heard is Keith Jarrett’s Köln Concert. I don’t normally like improv jazz, but this is a universe above anything I’ve ever heard before.
Here’s a couple I like:
http://www.youtube.com/watch?v=-vDG8MghMdQ
http://www.youtube.com/watch?v=_Sdb37nYIY8&feature=related
Renditions of “Auto-Tune the News” on piano. Suprisingly good.
In the spirit of Satie: Pleiades Dance from the japanese composer Yoshimatsu Takashi (this is only one of many Dances, quite a few links on youtube)
very simple, but hauntingly beautiful music
Philip Glass has quite a generous supply of stuff streaming at his site.
The piano music I most listen to is the 8 volumes of The Art Tatum Solo Masterpieces.
The other thing is Hélène Grimaud’s 2001 Rachmaninoff with Piano Concerto No. 2, Prelude Op. 32/12, Etudes-Tableaux Op. 33/1, 2 and 9, Variations on a Theme of Corelli. I heard the Prelude and the Etudes-Tableaux in Note by Note: The Making of Steinway L1037 I thought they were the best music in it. And she is great.
Bill Evans’ Conversations with Myself. It’s debatable whether or not it satisfies the conditions of the OP, since there are three instruments on it. However, since all three are Bill Evans playing piano, it might squeeze by 
It’s almost the right time of year for this: “Winter Wonderland” done boogie-woogie style by Harry Connick, Jr.
Perfectly righteous! We need us a thread for more of the “Christmas with a twist” sorts of things. Want to start it up, Lute Skywatcher?
Are you only looking for classical? George Winston has a beautiful line in what he describes as “rural folk piano”; it’s delicate, hypnotic, restful music, often inspired by natural landscapes.
I was given a tape of his by a friend many, many years ago, which had this piece on it, and I’ve never looked back since. http://www.youtube.com/watch?v=B-mLkzFUzTA