Legend of the Galactic Heroes is the best anime in existence. It is a Space Opera , based on several novels. 110 episodes.
[url=]http://en.wikipedia.org/wiki/Legend_of_the_galactic_heroes[/url]
It is not an insult to one's intelligence like most other anime. Infact it's quite stimulating and grandios.

I highly enjoyed a few anime series, but the two problems i found were that
(a) They are almost all the same. So after a while you realise you are watching the same 2 or 3 animes over and over.
(b) Most animes seem to run out of steam 2/3rd of the way through. They all start off casual, then start hinting at some interesting underlying story. Then end suddenly with an ending thats not half as interesting or satisfying as the one you expected. I've lost count of the number of series where that happened.
90% are pretty poor, to be honest. But right now anime is so big (particularly in the US) that some people just seem to be in love with all things anime/japan.

I've watched and enjoyed a few anime series:
Mai-Hime
Mai-Otome
Melancholy of Haruhi Suzumia
Ergo Proxy (aka. I've got no fucking clue what is going on, and why I'm watching this shit... I honestly have no idea of what exactly was going on in it)
Death Note
Elfen Lied
Blood +
Series I've only seen a few episodes off:
Nagasarete Airantou
Full Metal Alchemist
Neon Genesis Evangelion
Trinity Blood
And some others I can't remember right now.
I haven't really watched much anime since summer, apart from a few episodes of a couple of series that a friend of mine has showed me.
Haven't read much manga though. Some Starcraft and Warcraft related stuff, and some volumes of a series called Path of the Assassin.

I want to recommend One Piece :)
it is long running series. 396 episodes, 9 movies, few ovas (my OP folrder is 92GB big and i still miss around 15Gb worth of stuff). Last week it was ten years anniversary of anime.
It is simply amazing. Extremely funny, clever, with great characters (there is one who at first you don't like then you go: can he BE more awesome?) and unpredictable, amazing story.
After so many episodes I still cannot wait for another week to see next episode (especially after this week's episode. The madness just began)
of course i watch them subbed. One Piece was dubbed and simply slaughtered by fourkids.
The animation is goofy, and takes few episodes to get used to (especially since it is from 1999) but I recommend it with all my heart.

I never got into One Piece... we have Animax here on IPTV and they show that ALL the time. I just don't enjoy the weird style.

This one sounds interesting, I'll have to check it out some time.

A few that I can think of:
-Battle Royale, a series about a class of high school students forced to fight to the death in a government-sponsred blood sport. Interesting for the psychological interplay between characters, but very bloody.
-Death Note, another psychological police drama about a notebook that can kill people. A fairly long-running series, it does unfortunately seem to jump the shark a bit before it wraps up.
-Full Metal Panic, one of my favorite anime series, about an overly militaristic operative trying to fit into a normal high school and protect people from various enemies (usually piloting giant mechs). It's quite well-done, and has a good balance between plot and action. Plus, all three of the series (FMP 1, 2, and the comedic interlude Fumoffu) can be had as relatively affordable box sets.
-Full Metal Alchemist, another very good "serious" anime, about war, alchemy, and the dangers of playing god. I still have to watch the rest of it, but what I've seen so far has been quite interesting.
-They're fairly hard to get ahold of these days, but the comic series written by Johji Manabe (Caravan Kidd, Outlanders, Drakuun) are great if you like detailed artwork, epic, world-war-style storylines and strong female heroes.

I second everything in this post x1million!
I'm surprised nobody has mentioned Code Geass so far. That and Tengen Toppa Gurren Lagann are the only mecha shows I think are any good, but they are easily 2 of my favorite shows ever.
I really can't stand any of the mainstream crud like Naruto, Bleach, etc., but Full Metal Alchemist is actually pretty good.
Baccano is really awesome, Depression-era gangsters in America!
A couple really underrated animes are Voices of a Distant Star and Noein. A couple funny anime I'd recommend are Nodame Cantabile and School Rumble.
As for Manga, Berserk is awesome. Anyone who likes it should definitely check out Gantz, Battle Royale, and Gunnm (aka. Battle Angel Alita... yes, it's a lame title, but it's actually an incredibly sophisticated sci-fi manga).
IMO some amazing, but highly underrated manga are Blame, Yokohama Kaidaishi Kikou, and Karin.
It pains me to see ANYONE recommending Elfen Lied.

Voices of a Distant Star is grossly overrated. It was pretty, but not that interesting.

Noein is pretty good yeah, I bought it and didn't really like it at first, but the more I think about it the more I like it. As soon as I'm done with Gantz I'll probably watch Noein again.

Sorry for bumping this but it's the best place to say it =)
I have discovered a new favourite anime of mine: Tokyo Magnitude 8.0.
In short, it's ****ing brilliant. I actually caught the first episode by accident while flipping through channels a few weeks ago. The artwork is wonderfully realistic, especially during the opening scenes and the show boasts extensive research on the subject matter to add realism.
As the title suggests it concerns the aftermath of a magnitude 8.0 (Richter scale) earthquake hitting Tokyo and the resulting journey of Junior High schoolgirl and her younger brother as they try to get home.
Definitely well worth watching if you ever get the chance to.

Are there either giant robots or amorous tentacle monsters?

No. Only small robots in the first episode. Although you do get to see certain Tokyo based landmarks falling down on people...

Nana, Oh! My Goddess, Full Metal Alchemist and Nausicaä of the Valley of the Wind are my current favourite ongoing mangas. In anime front it's harder to choose as I watch so many, however anything produced by Ghibli goes above the rest.