For me, nothing is perfect. Not even perfect for "me", like based on my expectations and wishes and stuff. So a masterpiece can't be perfect, or there would be no masterpieces, which would sort of defeat the purpose of the word/this topic.
Took the words right out of my mouth. I'm not so good at designating the labels "perfect" or "flawless" to things, even if they come so close its painful. The flaws of works are what give them character and in and of themselves kind of define it along with the amazing aspects for me.
So I thought about it and the anime I can list in good conscious as amazing in so many ways:
Baccano!: Baccano is pure mind-stimulating crack of the absolute best kind. The classy yet crude 1930s setting and atmosphere, the crisp animation that can go from soft to crazy in a second, the piecemeal plot the jumps around every which way and gives just enough info to keep you at the edge of your seat at the key moments, but what really solidifies it as an all time favorite is that cast. Every character is not just insanely memorable, but contributes to the plot in some tangible way that brands itself into your brain. Just, my god, this writer knows what the hell he's doing (this series was originally a light novel). This anime only has 13 episodes covering the main plot (16 counting the afterstory epilogue that covers some loose threads) and in the span of that time he manages to flesh out, personalize and integrate the 17 named main characters in the opening, which doesn't even include the many others that show up and become a big deal as the episodes go on.
There's so much going on at once; action, ganster intrigue, unexpected romance, past conspiracies and supernatural occurances, yet it never felt like too much to handle, and I think this is due to the pervasive tone of morbid mockery of seriousness throughout. Mind you, there are tons of serious moments, but the series always has time to snap back and explore the more odd aspects of humanity through the unhinged characters. There's so many times where you feel like you shouldn't be laughing (I've never laughed so hard at a deranged psycho in a white suit dancing in a giant puddle of blood), but just can't help yourself with how ridiculous a scenario is portrayed.
This show is kind of infamous for having one of the best dubs of all time (silly accents, silly accents everywhere), so if you'd check it out, I'd recommend that track over subs any day (I mean, what could fit 1930's gangster america better?). Also, I really like its successor Durarara! too, but for me Durarara! doesn't stand up to the sheer force of Baccano!.
Princess Tutu: Probably one of the worst possible names for a title that could grace this page XD, but don't be fooled; beneath the frilly exterior lies a bastion of classical themes being ripped to shreds and rebuilt into something new. The show stars a young girl at a ballet school academy... wait a minute she's actually nothing but a duck... but wait she's actually a ballet princess in a book? The main character has two different alter-ego's, and the story kicks off when she meets and falls in love with and emotionless prince and vows to protect him. While this thread of the tale begins, the story behind the town she lives in is actually a lot more complex and sinister than what is initially let on. Princess Tutu is a series built on the themes of fairy tales; the main characters are actually proxies themselves of an in story author, who is attempting to control their every action and lead them to his desired tragic ending. Thus, in a struggle to find identity and purpose in their fate, they fight amongst each other, discover unexpected common ground, work together, find ways to step out of their given roles, and subvert the expectations of their creator (and sometimes even the viewer, depending on who you are).
The use of classical music in this series is brilliant (especially if you research the history of each track); every placement of a song in tandem with a ballet dance is a reference to some tale that ties to the events of the current plot. As for the characters, the group is fairly concise, with four mains (no spoilers) and a handful of recurring side characters. I will say that the mains (a set of them in particular) are some of my favorite characters from any anime ever, and the chemistry between the cast feels very natural (you can tell Ikuko Itoh, who also worked on the Sailor Moon anime, spent 8 years designing the main characters of Princess Tutu). The dub for this one is also excellent and I think it fits better, just because of the european background of the setting, stories, and soundtrack.
Oh speaking of Utena Thelazor, I've heard in many places Princess Tutu referred to as "Utena-lite", which makes sense since it shares some of the same themes, but is designed to be suitable for a younger audience (sort of...there's still lots of dark shit in there). Unfortunately I have no opinion on whether I would put Utena on this list or not because I haven't seen it yet (and need to, its been on my radar for a while)...
(Note: There was a manga adaptation of Princess Tutu but it sucks and completely misses every single point of the anime, don't read it)
Honorable Mentions: Time to cheat ! I probably shouldn't bother putting these here, since I don't consider them as quite measuring up to the level of the above and by virtue of me sticking them here, I'm kind of extending my boundaries just so I can ramble XD, but whatever. I mainly put them here because I had a few things to say about each in terms of anime genre's and conventions as a whole.
Cardcaptor Sakura: Fun fact; I'm not a huge fan of long runners. Most of the anime I've watched (and still gravitate to) are usually no more than 26 episodes, sometimes up to maybe 35 or 50 if I was feeling daring at the time (or the series is known for being that good). Cardcaptor Sakura is 76 episodes long (with two movies), but here's the thing; for the most part every episode counted. There was always something meaningful happening in an episode, whether it was getting another Clow Card power that could be used in a later fight, development between the characters (this series was great about character continuity, no bullshit resetting between episodes for the most part) or foreshadowing for a new plot development in the future. It still has one of the best slow build romances I've ever seen (between two 10 year olds no less), and really shows a thorough tale of coming of age from a young girl's point of view without being overly cheesy (like some other magical girl shows can fall into). Despite the mountains of titles CLAMP has created over the years, I think Cardcaptor Sakura remains their best.
Madoka Magica: Urrrrg, I love this anime, I really do, its great in so many ways. The eldritch world of the witches mixed with the exaggerated moe of the girls creates a fabulous contrast. The themes and relations to magical girl tropes are subtle yet potent at the same time, creating a tradegy you can't help but want to avoid. But it falls short because of one thing; the characters. Now, I really like the characters of Madoka Magica. They all have believable motivations, personalities, standout designs. It's really not about that. It's just that sometimes I feel like they end up being tragic constructs rather than fleshed-out individuals who formed their own motivations through shown experience. This is fine as a narrative tool. Using a character to make a statement is a legitimate method for storytelling. But for me, I like my characters to be the forefront of a story, and they have to be outstanding for me to rate it above and beyond.
Eyeshield 21: What HERESY I SAY. A freaking shonen sports manga has been spoken of in this sacred place XD? Well my boyfriend introduced me to this manga last year, and after about an hour of reading it, I was hooked hard. Much like Cardcaptor Sakura, every chapter of Eyeshield 21 has something in it that progresses whats happening; no staple shonen filler here. The cast is extremely fun (with several standouts that you'll be sure to remember) and bounce off of each other in so many ways (you know exactly what would happen if you put two characters in a room together). There is a protagonist (who has an excellent coming of age story of his own), but he doesn't suck up the spotlight at all. In fact, tons of other characters get equal amounts of screentime and their own arcs of improvement. There are no useless team members (everyone gets a moment to shine and make the reader see why they belong in the group) the games all play out like over-the-top shows of flashy special moves and play and are tons of excitement, but they almost always put them in the tactical context (with a few exceptions of character-driven level-ups) that makes sense and get somehow gets you super interested in the mechanics of football (or course a japanese manga would make me actually care about american football XD).
You know whats cool too? Because its a sport manga (and there's no actual threat of death), the main team actually loses sometimes and has to learn the hard way from it. It runs just long enough to not overstay its welcome and has a very satisfying ending (quite the achievement, considering how many great characters there were to speculate about). Overall, its a very well-written romp that still has that shonen charm. Highly recommend even if you don't like sports manga (which I didn't really before I read it).
(Do not watch the Eyeshield 21 anime if you can stomach any semblance of reading. It's a tragic insult to Murata's awesome art, with cheap ass animation that goes off model constantly. Despicable.)