Oog! Whenever I have a lot to say, I end up writing posts with terrible grammar and a hard-to-follow line of thought.
If any of that’s unclear, feel free to ask and I’ll try to rephrase it.
Wow, thanks for this. I checked out some preview clips of the GitShell soundtrack and it’s amazing! Downloading it now, worth every penny 
I’ll have to check out her other work!
Get a Yoko Kanno composition called “Tank!”, the theme song to the anime Cowboy Bebop. It is sheer awesomeness set to music (and happens to be the only anime I could ever give a toss about).
Going waaay back, there’s “Gigantor”, a Japanese cartoon I remember loving in the '60’s. Also from our friends across the Pacific came the hilariously awful “Johnny Sokko and his Giant Flying Robot”, which apparently had a budget of $1.75 per show. Watching the same evil henchmen (the Gargoyle Gang) get gunned down every week was a part of my regular viewing schedule.
Ah, good times. Probably not what you are after, though.
Oh, speaking of giant robot anime… one I just remembered, from an anthology called “Robot Carnival”. One of the segments is a hillarious piece with two giant robots fighting eachother in Tokyo… in 1899. Both robots are based on turn-of-the-century technology, including coal fired furnaces, ceramic tiles and wooden framework, ropes and pulleys controlling the limbs, and weapons in the form of an iron cannon and a battery of dragon rockets. At the end of the segment, Tokyo has been saved from the invader… and of course thouroughly destroyed in the process. 
Two ones that no one else has ever seen that I like are Kishin Corps and Orguss 002.
And of course you should check out Macross Plus.
There’s also some distinction between piloted mecha, which may be anthropomorphic, but are basically machines/extensions of the pilot, & AI robots.
Another vote for Escaflowne and Robotech. On the more lighthearted side is Martian Successor Nadesico, which follows a frycook / giant robot pilot who’s a giant-robot-anime fan himself. I thoroughly enjoyed it.
I enjoyed Giant Robo. Very retro feel.
The giant robots/vehicles in War of the Worlds were rather awe-inspiring, although probably not exactly what you’re looking for.
And X-Men 3 had the giant sentinel robots.
I remember “Johnny Sokko”, and also “Ultraman”, although I was never really sure what the heck Ultraman was. Robot? Giant humanoid of some sort?
Ultraman was an alien giant robot (going from memory), that can’t survive in our atmosphere. The human host crashed his plane into it, and they struck a deal of some sort. Ultraman was much cooler than Spectreman (on most occassions).
I think someone already mentioned Mazinger Z, which is what I was going to mention.
There’s a US anime type film out there with giant robot in it (I think it’s in the title). I can’t seem to remember it right now.
As for the core work in this genre, it’s been mentioned several times, so I’ll just add a me too: 1) Evangelion; 2) Robotech/Macross; 3) Full Metal Panic; and of course, the super-old Gundam (still very popular today, at least the figures are).
As for current games, the last time I was in England, my cousins and their friends were really big into Armored Core. The figures look really cool, but they’re small.
There was one in Lio yesterday.
Hmmm…I thought I had one, but on further research it turns out Goldiar wasn’t really a robot, but a humaniod something made of gold…still, one of the earliest “robot” looking shows I remember…Wiki link.
Video game wise, Zone of the Enders on PS2 had very nice big robot action… highly recommended. Havent played the sequal though.
One of the coolest, sleekest-looking giant robots of all time had to be Kronos
from the movie … you guessed it - “Kronos”.
That link has a synopsis of the entire movie with a generous amount of photos. Admittedly, the movie isn’t that good, but the robot and its “transportation mechanism” - it “walks” on vertically moving pistons - is really something to see.
If you are seeking out giant robot movies, this is a must see.
Ultraman is generally agreed not to be a robot, but an alien. Interestingly enough, Ultraman is one of Japan’s most popular cultural icons, with the franchise still going strong today. There have been numerous TV series, movies, and specials with the character.
The original series has been recently released to DVD in this country - it’s kind of fun to compare the American English dub with the original Japanese via subtitles. They did change a few things here and there…but not a whole lot really. Some of the plots do make much more sense in the original.
As far as Neon Genesis Evangelion goes, it may be easier to think of it as a Dating Sim disguised as a Giant Robot/psychological thriller/psedophilosophy/apocalyptic anime.
Heard The Iron Giant was good, haven’t seen it though.
Oh good times. Ultraman was on M=w=F and Johnny Socko T-Th. I used to know all the arm movements Giant Robot (yes, that was what they called him) had to do to get his various weapons to work.