These lucky anime titles have been viewed in their entirety by yours truly.

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Baccano! (TV) Masterpiece
Bubblegum Crisis (OAV) Masterpiece Cyberpunk at its best. Part Blade Runner, part Streets of Fire, the ultraviolent 80s retro-future world of MegaTokyo and the Knight Sabers makes for hours of entertainment. The music and strong voice performances are also to be commended.
Cowboy Bebop (TV) Masterpiece (dub & sub) A stellar series. Perhaps the most representative of what anime can accomplish, Cowboy Bebop is both entertaining and provocative. While it oozes "cool", it's not merely all style. It's a rousing jazzy ballad of the bizarre and the sorrowful. It's a must see for anyone.
Cowboy Bebop: The Movie Masterpiece (dub & sub) A great companion piece to the television series. Captures the flavor of Cowboy Bebop in a one-off that feels like a single episode type story but is on a grander scale that suits a movie. Fans of the series shouldn't miss it.
Fullmetal Alchemist (TV) Masterpiece A more enriched, layered variety of shonen, FMA is one of the highest quality titles of the decade, in both production values and storytelling. A large cast is balanced well, drama handled evenly, and despite some inappropriately timed comedy and adaptation missteps, it stands heads over most other titles of both its kind and not of its kind. Strong themes of brotherhood, abuse of power, and taking responsibility penetrate the surface, making this a more mature shonen experience.
Fullmetal Alchemist: Brotherhood (TV) Masterpiece At least as good as the first FMA series, Arakawa's wonderfully detailed world and colorful characters have a chance to shine in a more outwardly action-packed, manga-faithful series with some excellent production values. It's a great series everyone should see, in tandem with the first series.
Grave of the Fireflies (movie) Masterpiece (dub & sub) GotF isn't just a good anime movie. It's not even just one of the greatest. It's actually IMPORTANT to watch it. It's mandatory, as much or more as it is to watch Casablanca or Citizen Kane.
Legend of the Galactic Heroes (OAV) Masterpiece A grand masterpiece of Japanese animation. It's an impressive blend of politics, philosophy, and tactics examination, set to classical music. The cast is a who's who of voice talent. The space battles can be a bit dull, but any dullness is offset by the charismatic characters that take part in them. Truly an epic space opera. Not to be missed.
Lupin III: The Castle of Cagliostro (movie) Masterpiece (dub & sub) Highly overvalued as a Lupin movie, but a great example of Hayao Miyazaki's talent as a director of beautiful stories.
Macross Plus Movie Edition Masterpiece Macross Plus at it's strongest. The movie edition has a tighter narrative that doesn't stray and a few new visually appealing scenes as well.
Mobile Suit Gundam (TV) Masterpiece Yoshiyuki Tomino revolutionized the "real robot" branch of mecha with this 43-episode toy commercial/war drama. As it is with its age and purpose in the industry, the show is clearly dated and full of oddities of both the visual and directional, but it's still clearly a very mature take on the mecha genre and can be very gripping at times. "Mecha of the week" gets tiresome, but the movie trilogy fixes this.
Mobile Suit Gundam - The Movie Trilogy Masterpiece The original story of Mobile Suit Gundam, focused. A mandatory watch.
Mobile Suit Gundam: Char's Counterattack (movie) Masterpiece (dub & sub) The explosively epic climax to the battle between the Earth Federation and Zeon, Char's Counterattack has high-stakes action as well as Tomino's usual flair for the dramatic. Can one man change humankind? You'll almost wish he can.
Mobile Suit Zeta Gundam (TV) Masterpiece Zeta Gundam is the Gundam franchise at its strongest. Not only for its plot complexities and interesting characters, but because it addresses ideas and issues that most mecha shows don't as maturely as it does, or as well. And it has a dark, moody atmosphere that lends itself well to the story as well as the characters and situations.
Neon Genesis Evangelion (TV) Masterpiece (dub & sub) There's a certain attitude and approach to Evangelion which seperates it from the rest of the 90s mecha programs, and possibly a position within the great pantheon of anime greats. It may be, for all intents and purposes, the usual "fragile boy pilots father's robot" story, but character interaction seems to reveal a strong theme of (mis)communication and the mentioned "Hedgehog's Dilemma". It's generally well-directed, though production concerns and scene composition takes a bit of a drop within the last quarter of the series. While it can be said that it borders on pretentious, it's still a fantastic work. Plus, y'know, Pen Pen.
Neon Genesis Evangelion: The End of Evangelion (movie) Masterpiece Tang returns! And there's more of it than ever.
Now and Then, Here and There (TV) Masterpiece A beautiful, sometimes brooding, tale of the good and evil of humanity under extreme circumstances, NTHT is a simply animated, but complicatedly brilliant masterpiece of Japanese animation.
Royal Space Force - The Wings of Honnêamise (movie) Masterpiece A triumph of Japanese animation about the triumph of an increasingly decadent society's branch of dreamers and seekers. One man's journey for purpose will inspire and touch the viewer. It can be a tad dull at times but pays off in the end.
Spirited Away (movie) Masterpiece (dub & sub) My favorite Miyazaki movie, which seems to invoke a strange sense of unknown childhood nostalgia. Even though it's a fairly new movie, I somehow get a warm sense of familiarity, in a dreamlike sense. It moves me in that manner. Disney did a good job dubbing it.
Star Blazers (TV) Masterpiece A space opera for the ages, Space Battleship Yamato may show its age unflatteringly in its animation flubs and science fiction plot holes, but its approach to space travel and great direction inspires feelings of respect for this true anime classic.
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