If I had to choose only ten anime out of my huge collection (and I haven't actually got around to watching all of the series I've already bought, so this may change in the near future), I'd have to say that these are the best. I didn't include any movies, because you just can't match the depth of a series in only two hours.
Ah, My Goddess: This is the epitome of grade-school-level romances, a fact the show makes fun of more than once. Hapless college student, Keiichi's, incredibly good karma (due to his good nature always being taken advantage of) causes a beautiful, blond goddess to grant him one wish. Of course his wish is for her to stay with him on earth, but he doesn't consider all the trouble that will cause him.
My Bride Is A Mermaid: A junior high-school student ends up engaged to a beautiful young mermaid, who also happens to be the only child of the boss of the undersea yakuza. The only question is, will he survive once the hostile mer-mobsters take over his school in the guise of human teachers. This is the most laugh-out-loud funny anime I've ever seen.
Log Horizon: A bunch of World of Warcraft (or close enough) players get trapped in the game world and start to remake it in their own image. One thing I liked about this was the fact that the NPCs were all intelligent and had their own agendas, plus they outnumber the players enough to give their demands force.
The Melancholy Of Haruhi Suzumiya: A high-school student meets a strange classmate who disdains humans – she only wants to talk to aliens, psychics, or time travelers. Oddly enough, he discovers that all three are gathering around her, convinced that she has god-like powers and terrified at the consequences if she should discover them.
Demon King Daimao: A high-school student, planning to enter the priesthood, discovers that he's fated to become the demon king. Among those trying to either kill him or control his power are witches, werewolves, superheroes, demons, dragons, vampires, and pretty much everything else you can imagine. The series tries to pack too much story into twelve episodes, but it's still fun.
B Gata, H Kei: A high-school freshman decides to have sex with a hundred guys before she graduates, despite the fact that she's never even been out on a date. She picks one fairly innocuous classmate to start with. Needless to say, losing her virginity turns out to be more complicated than she expected.
Galaxy Angel: This is a long series of vignettes, set in the future, about the Galaxy Angel task force, a group of young women with unique training and expertise. It's pure zany comedy.
Genshiken: A slice of life series about otaku (geeks). It's nice to see people even geekier than me managing to make it through life.
Please Twins: An orphaned high-school student, barely managing to eke out a living on his own, is visited by two attractive young women who both believe themselves to be his sister. Only one can actually be related to him, a fact which becomes crucial when both fall in love with him.
Strike Witches: An alternate history WWII where aliens attack, drawing all the world powers away from war and into defending the Earth. The enemy is so powerful that only magic has any hope of defeating them – enter the Strike Witches, who fly around shooting the UFOs down with period weapons. Lots of fan service, but the story was excellent and the aliens were the most ALIEN of any fiction I've ever seen, to the point where you couldn't even predict what they'd shoot at in combat. (In the first naval battle, they spend much of their time shooting the ocean.)
C Control is basically an arena combat anime, but it's an insidious sort of combat. The fights happen in a strange reflection of the city called the financial district, and the fighters can't be injured. When one of them gets hit, he literally bleeds money and the one who loses the most, when time runs out, loses the fight.
The twist is, the money you lose is loaned to you by the proprietor of the district, with your future as collateral. Lose the fight and you lose a piece of your future appropriate to the amount of money you lost. One guy goes bust and, when he gets home, finds that his three children never existed. Another loses a little and fails a test. Still another loses by just as narrow a margin and finds that his vaccine didn't save some people who should have lived... but it didn't affect his future much.
Of course, no one can force you to fight, but every fight you skip costs you half of your personal fortune, and the future that goes with it. All the while, black money, indistinguishable from the normal stuff except to the fighters, is pouring out of the financial district and into our economy, and every time a major fight occurs, changes happen in the real world, usually unnoticed by anyone except those involved.
My entries for the "secular alternatives to the ten commandments":
See things as they are. Try to set aside bias, prejudice, and unnecessary assumptions. The universe is full of surprises—unless you've already decided that it isn't.
Know yourself. You are a remarkable creature. Never cease to be amazed by all the things you are and that you can do. On the other hand, remember that the rest of the universe is also amazing.
Explore. Every person, every creature, every place and time have a story to tell. Ask questions, look it up, or find the answers first-hand and tell others. Learning never gets old.
Don't cling to possessions. People are more important than things.
Be free of tradition. Listen to the advice and traditions you come across, but decide for yourself what makes sense. Never follow a belief or a person blindly.
Don't worry. Do what you can to overcome the challenges you face, but don't dwell on them or fret over things you can't change. Especially, don't worry about what people think of you. Do what's right and accept that some just won't understand.
Love everyone. Don't try to limit the best feeling of all. Love as much as you can stand to. Everyone is worthwhile.
Don't carry a weapon unless absolutely necessary. When you expect a fight, you're apt to find one. When you're unarmed, you're less likely to take foolish chances. Use your head, either way.