For my fellow otaku...
Dragon Ball franchise - Never really cared for it due to having the ugliest character designs I have ever seen in anime, but you can't deny that it was one of the quintessential series of the 90s. The modern "shonen" genre really wouldn't exist without it.
Sailor Moon - Another quintessential 90s anime, and the definitive magical girl anime. This one I actually tried to get into, but I couldn't stand it due to its monster-of-the-week formula, repetitive transformation scenes and the atrocious job the English dubbing company gave it.
Pokemon - Animated tie-in of the game that gave birth to an absolute behemoth of a franchise. Unlike Sailor Moon or DB, I was actually a huge fan of Pokemon back in the day, and I am strongly nostalgic over the first couple of seasons as well as the first couple of games. This series is still so well-liked that other people who grew up in Generation Y think I'm the weird one for having outgrown it!
Trigun, Cowboy Bebop and Big O - Anime that were arguably more popular in the US than they ever were in their home country of Japan, due to utilizing tropes and themes that appeal to Western audiences (Trigun = Western, Cowboy Bebop = renegade outlaws, Big O = film noir). I've seen a couple of episodes of each show but they aren't my cup of tea.
Neon Genesis Evangelion - One of the few anime that I've watched in its entirety, including the movie finale. Knowing the spoilers beforehand did nothing to prevent my mental and emotional state from being put through the wringer. You either love it or you hate it. I personally enjoyed it and can understand why it's widely considered a classic, though I'm not sure if I'll ever be able to stomach giving it a second go-around.
Elfen Lied - It's as famous for its shock value as it is for its intricate study of human nature. Makes Evangelion look like Hello Kitty by comparison, with its extreme brutality. It's only 12 episodes long, but I have never been able to make it past episode 5.
Full Metal Alchemist/Brotherhood - One of the most popular and well-loved anime series, it endures continuing popularity. I have seen a few episodes and can see why people consider it a classic - it has deep characters and an emotional, engaging story. I know some of the spoilers (including how it ends) but I do plan on finishing this series one day.
Revolutionary Girl Utena - A cult classic, often considered to be the shojo equivalent to Evangelion, not to mention a flagship yuri series. The themes presented in the story struck me in such a way that they still have a influence on my art and imagination to this day. I liked the first season but became unsure if I could finish it when the second season became noticeably darker. I quit while I was ahead, not wanting to put myself through another Evangelion-style mindscrew.
Puella Magi Madoka Magica - Basically, what Evangelion is to mecha, this series is to magical girls. Also like Evangelion, one of the very few anime that I have actually finished. Famous for its shocking plot twists as well as its ultra-moe characters (who suffer terribly as the series progresses).
Attack On Titan - According to most otaku, this is the must-watch anime of the modern day; possibly a classic in the making.. I gave it a sincere try, but to be honest it really doesn't appeal to me.
That's all I can think of for now. It's getting a bit late.