-Battletech Universe, namely the Clan Invasion and the FedCom Civil War. You're gunna see a lot of anime in this thread but I don't think a lot of people are gunna even know about the Battletech mythos.
-Universal Century Gundam as a whole
-Getter Robo, namely Shin Getter Robo Armageddon. That intro is more powerful than the whole Gurren Lagann soundtrack.
-Pacific Rim but to be fair not really for the story.
-actually don't have a fifth, huh... Now I feel kinda bad.
Basquash: Loved the story and presentation for the most part, the robot design took some getting use to at first but eventually grew on me.
Metal Gear Solid: Rex was my favourite of the Metal Gears, the story was also great.
Code Geass: While I liked the story the robots werent really anything special.
Gurren Lagan: The robot design took some getting use to and I didnt really like the story except for the season climaxes.
Voices of a Distant Star: was an enjoyable movie and the only other mecha thing I can remember having seen at the moment.
If i cant use games then Pacific Rim was pretty cool too.
Honorable mentions:
Mechwarrior 2 had some great mech designs but I couldnt tell you the story if my life depended on it.
Neon Genesis Evangelion had a great story (particularly the manga and films) but I didnt put it on my list because (SPOILER)theyre technically not "robots" since eva's are made of people
Kefka's coming, look intimidating!
Have a nice day!!