Ok I'm going to be real. A year ago I decided to torrent a shit ton of anime (500gigs) so I have a decent gauge of whats good.
First off Death Note is not the best anime, to me its just good. The reasoning? The 2nd half of the anime. They introduce some characters with no backstory and it spirals out of control.
These are my recommendation
Code Geass season 1 (season 2 isn't bad but season 1 is 100% better)
Gundam 00 Season 1 and Season 2
Magical Girl Madoka. DO NOT LET THE MIDDLE SCHOOL GIRLS FOOL YOU. This anime is voted 2011 best anime and its dark as fuck. Its only 12 episodes go watch up to episode 3 before you judge.
Baccano (its like durarara but only a metric tons better. Better char, better plot, and better pacing)
Bakemonogatari
Ghost in the Shell Stand Alone Complex season 1 and 2 (season 2 is better overall)
Gurren Laggan (one of the best "epic" in recent anime, it truly feels like its telling an adventure from start to finish)
Full Metal Alchemist Brotherhood. Don't watch the shitty first version, this one is legit. It sticks to the manga, its dark as fuck and characters don't flip flop personalities.
The Girl who leapt through time.
Eden of the East
Katanagatari (its a tribute to old school samura/ninja style genre. None of this Naruto bullshit)
Gosick (a darker tone anime shown as like a mystery novel)
I HIGHLY recommend Madoka. I mean at first the shit will seem pedo-ish due to the age of the main character, but that shit changes so fast after episode 3 and then its obvious why the age of the main characters are what they are.
Also Code Geass Season 1 is still amazing. Season 2 is like "Wtf? ...ok i guess?"
edit
Most of the anime if not all i mention has little to no fillers.
edit 2
code geeass fan made trailer without that many spoilers. Its really a good anime, and it has some elements of Death Note. The only issue I have with it is that it has the same problem as death note but to a lesser extent.