Code Geass has been (and still is) accused of being an anti-American Author Tract. When asked about the subject, director/co-creator Goro Taniguchi's response was "I know some authors have political messages in their works, but that wasn't my intention; I just wanted to tell an entertaining story." Later, when asked again, he responded "You mean America and Britannia are exactly alike? I had no idea!" In case anyone's confused, it wasn't a "Well, duh" statement — it was more along the lines of: "The US is currently led by an Emperor with WTF-curls who believes that all men are not created equal?"
Earth Maiden Arjuna starts out as a fast-paced mature Magical Girl series. Then it quickly veers into very heavy-handed ecological preaching. Tolerable, because the animation is freaking sweet, because Theresa is really badass, and because Juna's transformation is damn cool, but the storyline is still Anvilicious to the point of being distracting, and full to the brim of very bad science about why Science Is Bad.
Fairy Tail wants to remind you that friendship is powerful and good. And that your friends are important. And that they make you strong. And you can't lose with friends at your side. And that even the impossible is possible with friendship. etc. Nearly every battle has at least one character proclaiming this, and sometimes even pointing out that the reason that the villain is losing is that they lack such friends. Note that while it is extremely common for Shounen series to preach the importance and power of friendship (and Fairy Tail in particular can get away with the fact powerful feelings like friendship can actually fuel magic), both fans and detractors of Fairy Tail can agree that the series takes it a little too far sometimes and it's almost like Mashima doesn't want us to forget that Fairy Tail is made of these good friends.
You soldiers can decide to live and die by any rules you want, commandant. You can play any games you want, but civilians shouldn't have to lose their lives as a result.
Incidentally, most of this came about of it being based on World War II.
The series created in the 2000s (including Gundam SEED, Gundam 00, and Gundam AGE) want you to know that war will end when everyone understands each other.
Another Shoji Kawamori piece, Macross Zero, mixes spectacular mecha battles with the seemingly-opposite message that all warfare is inherently evil. It's set on an island that's a mostly-primitive Eden, inhabited by innocents. The shaman/priestess freaks out over the arrival of UN forces to defend the island, saying they're possessed by evil spirits that are prophesied to destroy everything. For the first half, this is played as "silly superstitious witch doctor". But by the end, you realize that she's absolutely right. The island paradise gets tac-nuked into a wasteland, and only her Heroic Sacrifice keeps the entire world from being obliterated.
Technically, Super Dimension Fortress Macross shows that she was only half-right. It was the alien Zentradi who wound up destroying most of the Earth and its people, and it's human culture that leads to the end of the war.
Masashi Kishimoto, author of Naruto really, really wants you to know that revenge is bad, kids. Also, that friendship, forgiveness, and self-sacrifice will solve anything.
Having been inspired by its creator's battle with depression, Neon Genesis Evangelion (particularly the ending — both of them) contains numerous sequences containing in-depth discussions of the human condition and concludes with a lengthy expose on the thought process that leads the main character to overcome his own depression, go on living and reject the Assimilation Plot he finds himself a part of.
Team Medical Dragon was written by Akira Nagai, a practicing doctor — and the manga basically centers around a maverick (but exceedingly skilled) cardiac surgeon and his team fighting against bureaucracy and corruption in the Japanese health services. It's particularly jarring when you realize that all the protagonists are incredibly good-looking compared to most of the antagonists, who are practically caricatures.
The issue with the looks is somewhat taken care of in the live-action version, with the antagonists having a fair amount of attractive people, and Dr. Asada being the only one pointed out to be good-looking.
Most of Hayao Miyazaki's movies have at least one segment that preaches the importance of respecting and preserving nature. That is if the plot itself isn't completely built around the aesop. Miyazaki often protests that he does not make films with the intent of sending messages, he just makes them to entertain and for profit. Fans have a hard time believing that given his criticism about capitalism and globalization.
Additionally, a few of his films contain an anti-war message, which makes sense considering he grew up in the 1940s.
Osamu Tezuka did this occasionally. In Black Jack, Tezuka often criticizes the current state of the medical establishment, lent some weight by the fact that he was trained as a doctor before becoming a manga artist. His science fiction stories, including Astro Boy often discuss the dehumanizing effects of modern society technology, but counterpoint it by showing all the good that can come of modern technology. Karma, the 4th (or 5th, depending on the localization) volume of Phoenix series is largely built around Buddhist themes, discussing Karma and reincarnation at length and lamenting the corruption of the Buddhist faith by political interests. The later (and sadly, final) Phoenix story Sun does something similar with Shinto.
Tezuka's science fiction book Apollo's Song did the same as Astro Boy, but touched on the nature of love and romance (not to mention Greek Mythology) as well.
Some of his stories that focus on nature like Kimba the White Lion tend to have a Green Aesop, but Tezuka tends to make it play back-burner to other Aesops about family and sacrifice.
Yamada-kun and the Seven Witches can sometimes be a bit heavy-handed with its You Are Not Alone message, to the point that the misery of most characters doesn't last more than a few panels until they see a helping hand reaching out, usually due to the titular protagonist's Chronic Hero Syndrome. Surprisingly, the series' message about the value of friendship isn't that heavy-handed (though still present), given that Yamada tends to befriend the people he helps after he helps them, so the Central Theme isn't really about helping your friends as much as it's about helping people in need.
The manga Yu-Gi-Oh! has the main theme of friendship - namely, that it can overcome anything and is better than working alone. This is all well and good, except that Kaiba insists on working alone and has achieved quite a lot for it - namely, his own company and is able to provide for his little brother. Situations where friends can be poisonous don't tend to be shown, and while support makes it easier to win with high stakes, players in real life can't give one another hints. Similarly, manga-ka Takahashi said in an interview that he believed Jonouchi / Joey's casual attitude towards games was stronger than the philosophies of the other characters, which makes sense more in real life than in Yu-Gi-Oh! where losing a game can actually cause people to die.