I'm out of the loop on newest anime, but the following are golden.
MOBILE SUIT GUNDAM
It's very old compared to just about anything (that I've noticed) listed here, but check out the original Gundam. It isn't for everyone, especially if you're looking for the kind of over-the-top, you've activated my trap, but I knew about your trap twenty episodes ago, but I have a secret form kind of anime that you find everywhere. It isn't even like the more recent Gundam anime, which are full of poorly written angst and melodrama. It's a serious, grim kind of anime with more subdued characters who deal with the situation at hand with a greater sense of gravity and realism. There aren't any pop idols who double as war-fighting, grand politicians. There aren't any obnoxiously large breasted women running around with lasers and Sentai uniforms. There isn't a bathhouse episdoes where the guys try to sneak a peek, or one of them end up trapped in the female bath and has to avoid detection.
It's without all of the tropes and bullshit that a long-time viewer of anime might grow tired of running into. It's seinen at it's best, and manages to throw in some believable and fitting levity to keep things from being too dark.
It's only downside is that the animation is from the late seventies, having aired in 1979. The quality of the art and animation can be spotty at times, but it's generally better than most of what you'd find in televised animation anywhere in the world during the period. It's one of the most iconic anime in Japan, and is considered the father of the real robo genre. It's definitely worth checking out, but I suggest watching the movie versions. There are three in total that condense the series into a superior package.
BECK: MONGOLIAN CHOP SQUAD
Do you like music? Then you need to go watch Beck: Mongolian Chop Squad. Why? I have yet to see an anime that does the band genre better than Beck. Now, I know it may be a niche genre with slim pickings, but Beck is really good.
First off, the music is actually pretty good. I'm not a fan of most Japanese rock, however what you have in Beck is some decent stuff. The real-life Beat Crusaders were the driving musical force behind the anime, so it has an authentic feel. The music isn't just some studio band hired by the production company to belt out some quick tracks. The songs in Beck can stand on their own feet, and still retain the raw, amateurish feel of a band just starting out. The only problem with the music is that they cannot possibly live up to the bar the author set with the main character's voice. This creates a bit of a disconnect, as you watch the fan reaction on screen, and then gauge it against your own more realistic response. Another cool thing is the English dubbing of the series; they actually re-recorded the music in English. I think it's really cool that the localization bothered to do this, however the results are mixed. When translated into English, the rapcore songs are a bit lost. Chiba, the hip-hop/rap front-man, loses a whole lot of his aggression and flow. I also found that the songs themselves feel a little more flat.
The animation isn't anything special. It's the typical quality you'd expect from most anime productions. It isn't bad, but it isn't OVA quality or anything. However, the feel of the manga is preserved in the art style and animation, so it's a boon to the fans of the manga. Both the opening and ending credits are done very well. Both capture the feel of the anime very well. The visuals are satisfying, and the music is just fantastic.
Where this anime really shines is the story. Beck is a surprisingly realistic tale of what it's like to get a band going. While this is a tale of a shy novice realizing his potential and talent, he doesn't get there overnight. Yukio "Koyuki" Tanaka, can sing his ass off, but he's a diamond in the rough. For what talent he does have vocally, he lacks it elsewhere, as he picks up a guitar and learns to play. For months he practices, and for years he performs, but it isn't for at least four years that he starts to come into his own. While the manga does extend years after the anime, I assure you that Koyuki isn't portrayed as some prodigy god. He doesn't give a lopsided smirk, flick his nose with his thumb, and then rattle off a shonen one-liner. He's a humble guy, and he knows he isn't great, but he diligently works harder to get better. He's written more like a real person. Throughout the series, the band deals with chemistry issues, disappointing performances, and the brutal expenses of a tour. Things go wrong for Beck more often than they go right. However, when things do go well, when the stars are perfectly aligned, their triumphs are great. The hard work and passion they put into their music pays off. It's very satisfying to watch or read.
So, go watch Beck! You won't be disappointed; It has action, comedy, romance, and awesome music!
AKIRA
I'm not sure if I even have to explain why anime fans should check this one out. It's one of the most influential anime of all time, and a serious milestone in Japanese animation. It's beautiful to look at, the story is great, and it's all around amazing.