My first topic. Hope this hasn't been done before. These are some of my favorites:

Three Clouds in the party: http://www.youtube.com/watch?v=FPrP9IxcnG4

Level 5 Tifa without materia and with her original equipment beats the Midgar Zolom: http://www.youtube.com/watch?v=h31_Wk9A2Io

A guy who is very good at the snowboard minigame: http://www.youtube.com/watch?v=nu5oMpUEkvo

Level 2 Yuffie beats Emerald Weapon: http://www.youtube.com/watch?v=CWBfeBTGgE4
Part 2: http://www.youtube.com/watch?v=YWd38aVtlyI

Aeris vs Sephiroth (final battle): http://www.youtube.com/watch?v=j4xL-AretME

The game totally f's up on the way down to Jenova: http://www.youtube.com/watch?v=qhmv_IJLVCY

Sephiroth as the main character: http://www.youtube.com/watch?v=wtpDlNtZ5NE&NR=1

What are your favorites?