I'm gonna be a big old Genwunner and say Blue/Trace's theme and Lance/Red's theme. It's incredible how intimidating they both sounded even back in the era of 4-bit/8-bit music. I know Blue's theme in particular has been remixed to death already, but that doesn't make it any less great.
I have to admit, though, that Leon's theme made me choke up as well because of the sense of finality it conveys, with what the callbacks to the classic Hall of Fame theme and the audience cheering for you.