I'll go for the lesser-known
Vision of Escaflowne.
A fantasy epic from way back in 1996, it remains my favorite anime of all time. I thought it was complex, original, and deep, without being pretentious - everything about the series was just well-executed. I would recommend it to everyone, with the caveat that the first episode is seriously misleading - the series is
NOT about life and love as a high school girl in Japan. The protagonist happens to be a high school girl in Japan - which is a turnoff for some, since mecha anime is generally supposed to be butch and testosterone-dominated - but the story shifts gears soon enough.
It's not exactly a light-hearted anime, either, so don't go in expecting that.
If there is a downside, though ... the mecha aren't necessarily the coolest I've ever seen. If we're talking about the coolness of the big robots alone, then
Escaflowne might not cut it.
These guys are pretty cool, though:
They're the villains of the series, and those five barrels can shoot both flame and liquid metal, which can be hardened at will into any shape by the pilot. They can also fly and some can turn invisible.
If they sound a little overpowered - well yeah. Much of the series involves the bad guys kicking the good guys' asses. Like I said, it's not that light-hearted.
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BTW, there are two versions of Escaflowne out there. The first is the series, which is excellent. The second is the movie that was released some time after, which takes the bold route of not really having anything but aesthetic similarities to the original. You have some characters with the same names and similar looks, but dramatically different everything else.
Also, the movie sucks. Do not watch the movie.
Unless you hate the series (in which case you deserve to suffer), in which case, please watch the movie as penance.
(edit 2)
Also,
Escaflowne was released for US broadcast on Fox some years back, but it was substantially re-edited, presumably to remove the more gory stuff and make the series more broadly appealing.
Don't watch the US edited version. Unless you enjoy misery and mild confusion.