"Emotional" Piano Music?

Never meant to belong
Bleach OST, one of my personal favourites

I have to add one more: How about "Farewell to the Piano" from the Battle Royale 2 film, it plays very emotional together with the scenes.
Enjoy: https://www.youtube.com/watch?v=pmqZoQcoE9k

And maybe you can edit your first thread and add a list of all suggestions? So we can all cry when we need to!

My absolute favourite piano pieces are by Erik Satie, a French minimalist composer. They're called Gymnopédies, and there are three "movements", so to speak. They never fail to evoke a strong emotional reaction from me, and I love playing them as well.
Moving from the classical (even though Satie breaks a lot of norms of traditional classical music already), one of the most beautiful modern solo piano compositions, in my opinion, is Aphex Twin's Avril 14th.
And, of course, since I really can't discard these: La valse d'Amelie, Comptine d'un autre été: L'après-midi (both from Yann Tiersen's fantastic soundtrack for Le Fabuleux Destin d'Amélie Poulain), Ludovico Einaudi's Oltremare, Philip Glass' Opening from Glassworks and, so far, my favourite modern entirely piano-driven soundtrack, Her, by Will Butler from Arcade Fire and Owen Pallett, from which I selected the two songs Song On The Beach, and Photograph.

Worth mentioning also is Nine Inch Nails' Leaving Hope, GoGo Penguin's Murmuration, the very simple but very powerful Radiohead's Daydreaming, Bent Knee's Untitled and BADBADNOTGOOD's medley Title Theme/Saria's Song/Song of Storms, probably my favourite renditions of those Legend of Zelda songs. These aren't exclusively piano, mind you, but are mainly piano-driven, with some prominent organ on the latter as well.

Besides "River flows in you" which was posted by someone already you can try these:
14th Song - D. Grayman
Lead Me out of the Dark - Crown the Empire
Le fabuleux destin d’Amélie Poulain - OST

Oh and also here are some great artworks with a stunningly beautiful video content!
The Mountain
The Water
The Arctic Light
The Aurora

Other Lives - Black Tables

HUGE LIST from my bookmarks! :D

Joe Hisaishi - Summer
Steve Barakatt - Rainbow Bridge
Claude bolling - Irlandaise
Steve Barakatt - Flying
David Lanz - Cristofori's Dream
Yiruma - River Flows in You
OYSTEIN SEVAG - Crystal Palace
David Lanz - Return To The Heart
YURIKO NAKAMURA - LONG LONG AGO
GIOVANNI ALLEVI - Go with the flow
Kim Kwang Min & Yiruma - The Way To School
David Lanz - That Smile
Richard Clayderman - Ballade pour Adeline
Claude Bolling, Jean-Pierre Rampal - Sentimentale
Ryuichi Sakamoto - Merry Christmas Mr Lawrence
Ludovico Einaudi - Una Mattina
Brian Crain - A Walk in the Forest
DJ Okawari - Flower Dance

Mostly because I know the stories with these songs:

FFIX - Melodies of Life
FFIX - You're Not Alone

I'll add more later.