Let's see, "has effect" and "rewatchable". ...
Hajime no Ippo. A combo sports (boxing) and slice of life show. Every character is good to great, the story is very down to earth (the voice of Rock Lee is the main character too) and it's so watchable, so many times. And, it's still going. The latest season just concluded on Crunchyroll.
Samurai X, Trust and Betrayal. A not-very-well edited movie, but a great story. I just rewatched it again and it was just as good as I remembered. (I recommend Remembrance for its fantastic fight scenes only, because it includes flashbacks to the made-for-kiddies TV series and that kind of leaves you lost. Worth watching the series too, but it is a kid show fyi).
Black Blood Brothers. A rather short and somewhat borderline-young selection here, but the story is full-adult, quite vampire-noir, completely original and very much a steamroller that will take you straight through from the first episode to the last if you aren't careful
It also has great characters. a great plot twist at the end, and omg, best female villain ever. "Endless jubilation and endless lamentation. It's all ahead of us" indeed. I seriously wish they'd gotten a series going from it.
Highly recommend Last Exile too, as others have. You cannot not watch that show. It's exactly what makes anime an art form that can hold its head high right next to anything that's ever come out of Hollywood. The intro is itself a terrific AMV too.
Avatar the Last Airbender is also a slightly "young" selection, but I found it completely watchable as an adult with my friend's kids. It's kind of a Disney-class anime in that respect - very high production values, epic story, great characters, and anyone can watch it.
Try Claymore if you like medieval horror and a tense, creepy story. It has some action editing issues like excessive action tropes (long bullet-time, talky in the middle of fighty, excess last words, etc.). However, I really recommend the story up until the end of the battle for Pieta, where they depart from the manga story and create an artificial finisher (the manga is several times as long now). The finisher isn't bad, but it isn't the same flavor as the story. Still a good watch and I own it, so there's that.
FYI Full Metal Alchemist has two different flavors - the "original" and "Brotherhood". Both are somewhat hard starts, they both do the "in medias res" thing of not telling you the backstory until some episodes have passed (a huge mistake in my book) but once you have gotten past those episodes and put things together in your head, it's one of the best shows ever made. I actually liked the first more, but "Brotherhood" sticks far closer to the original material and has a better ending, so flip a coin there.
Other than that, the "classics" like Ghost in the Shell:SAC (the first one) and Cowboy Bebop are always recommended.
Best of luck whatever you choose!
