1. Attack on Titan
Hugely popular in the United States, Attack on Titan is a story of some tough teens dealing with the stresses of life in a world where gigantic humanoid monsters keep wrecking shit. With high energy, action, adrenaline, and lots of cliffhanger episodes, it's easy to see why this show quickly became incredibly successful.
One thing that's cool about this show for me is the Titans themselves. They're so mysterious and strange that almost everything the human characters learn about them raises more questions than it answers. Who made them, and for what purpose? Why can't they reproduce sexually? Do they reproduce some other way? Are they natural or man-made in origin? Why are some people able to change into Titans? Why are there different types of Titans? Etc. Basically, I was upset with this for not answering or explaining more, but I'll give it props for centering its conflict around such an interesting mystery. It's also very thrilling to watch.
2.Fairy Tail
Like how Fullmetal Alchemist draws inspiration from real Western philosophy and politics, Fairy Tail draws inspiration from Western storytelling, folklore, astrology, and so on. And it has some unique elements, like a blue cat with wings, that are just fun and part of the unique lore of this show. The main characters are Lucy, a newb in the famous/infamous guild of wizards called Fairy Tail, and Natsu, a dragon-boy wizard who is fiercely loyal to the guild. This show combines European fantasy concepts like dragons with typical shounen stuff (like the protagonist eats a lot, likes to fight, yells, and sometimes loses his temper, what a shock!). It has some very amazing characters, and the animation is very nicely done. I love the character designs and the art's use of color. But ultimately, this show is not just a pretty face, it's about the bond of loyalty shared by all the members of the guild, so it has many moments that are heartwarming and touching.
3.Inuyasha
This one used to be a late-night cable favorite for me for many years. Inuyasha is the story of schoolgirl Kagome, who travels back in time from our present to medieval Japan, and finds the titular half-human dog demon Inuyasha pinned to a tree by an arrow. She rescues him, learns about his past, his struggles with the annoying plot token known as the shikon jewel, and her role in all of this, as she is the reincarnation of a priestess named Kikyo, the woman who originally pinned Inuyasha to the tree. Yes, this show has its flaws, mostly in that there are vast oceans of filler in between actual episodes where the plot even budges an inch. But, like Fairy Tale,Inuyasha is entertaining, heartwarming, and immensely popular in the US.
4.Ghost in the Shell
While Stand Alone Complex the anime series was a lot of fun for me when I watched it on Adult Swim back in my day, the Ghost in the Shell movies are also breathtaking, and I've watched a few episodes of the newer prequel series called Arise and liked that as well. Ghost in the Shell is a major (no pun intended) work of science fiction anime. I often use it as a recommendation for people who don't like anime but who might like related science fiction works. The first move, for example, borrows from Blade Runner, a sci-fi classic. As can be expected from a work of the cyberpunk genre, Ghost in the Shell explores potential conflicts caused by potential advances in technology. While some types of sci-fi, like Star Trek, believe that technological progress will result in social progress, cyberpunk anime, literature, movies, video games, etc. have a more pessimistic viewpoint, arguing that there is a dark side to human nature that will not go away just because of improvement to technology.