In no particular order, the following are some of my favorite anime shows. These are all completed series I have fully watched, sometimes multiple times.
Avatar: The Last Airbender
This is the series that really got me into watching anime. A fun story about the last airbender traveling through different lands and ultimately defeating the evil king in the fire nation. Simple, children’s cartoon with excellent characters and relatively predictable plotlines, but very well executed.
Death Note
This is one of the first anime I watched. The premise got me hooked immediately. A shinigami drops a diary upon the earth. The diary is special: anyone who’s name is written on it dies. A high school student finds the Death Note and immediately starts to build a world reflecting his ethics. The ending drags on for a little, but an overall gripping story.
Assassination Classroom
Another weird premise: An octopus monster has blown up the moon and threatens to do so to the earth unless he is allowed to teach Class 3E of Kunugigaoka Junior High School. The monster turns out to be very caring. He teaches the kids, among other things, how to be top assassins so that they can kill him before the term ends. A surprisingly wholesome show with an emotional ending.
Castlevania
A vampire hunter, a mage, and Dracula’s son endeavor to stop Dracula’s genocide of the human race. Perhaps the most complex characterization of Dracula. Great storytelling, fine fight choreography, and some of the sharpest, wittiest dialogues I’ve seen in any anime. The show drags on for a season too long; nevertheless it remains engaging throughout.
Steins Gate
The best time travel story. The MC and his friend build a time machine, which causes them to get involved in an intricate plot where every little action has unwanted knock-on effects. The MC has to go on a seemingly hopeless journey across multiple timelines to save his childhood firend, his girlfriend, and the world. Watch both Steins Gate and Steins Gate: Zero. The latter gives new meaning to the last few episodes of the former.