The form doesn't allow as much depth, in the sense that even a 52-episode series can only cover a short novel. But I've found some very thought-provoking works (as, to be fair, I have in comics); particularly:

• Kaiba is my go-to example of something that would be impossible in any other medium; it's a story about personal identity in a world where memories can be transferred between bodies, and uses this blobby, amorphous style to fit with that - everything's strange but semi-familiar, the same way it is to the characters in the story.
• The Melancholy of Haruhi Suzumiya - a story about the teenager's gradual realization that you're not the center of the universe, but rather one ordinary person among many, refracted through the prism of a clever premise. Has this hyper-contextual style where everything is a cultural reference, but without ever seeming pretentious (perhaps because of the exceedingly wide range). Of course you can simply read the novels, undermining the argument somewhat, but the adaptation has some clever visual tricks and the immediacy is valuable. Its stock has fallen in recent years due to a very viewer-hostile approach to the second season; I'd advise only watching the first season, and definitely watch in broadcast order rather than DVD/chronological order (the baseball episode is the third "proper" episode).
• Haibane Renmei - About personal identity and finding your place in the world, or perhaps living a life decided for you. Very much in the language of mystical symbolism, and rather a love it/hate it case, but I found it interesting, and it would usually push me into a reflective, detached state.
• Kino's Travels - a series of short stories. Like the best episodes of Star Trek, really shows you a society that's mostly like our own, but with some crucial difference. I can't compare it to the novels as I haven't read them, but I find with this kind of worldbuilding the immersion of a visual medium can be valuable.

I could name plenty more shows that were thought-provoking stories (Psycho-Pass and Eden of the East come immediately to mind), but these are the ones where the animated form seemed to really matter. (Though to be honest I'm surprised you list Death Note - I wouldn't call it especially thought-provoking, and it was a manga before it was an anime).