My backlog is big enough that I really shouldn't, but every other week I'll rewatch certain episodes of series I really liked.
Rewatching entire series is just not doable for me though.

A backlog is a fun thing to experience. It should not be something that takes over your time and demands you watch it so that you can't watch something else you like.

Seriously. Tell me you never rewatch all of Dog Days.

The only things of DD I sometimes watch are my favorite battles.
Season 1 the final struggles in terms of plot.
Season 2 the Nanami-Brioche fight and the Eclaire-Leonmichelle fight.

Some other stuff:
Naruto:
Rock Lee vs Gaara
Sound five battles.
Chiyo/Sakura vs Sasori
Naruto (4-tails) vs Orochimaru
Hidan/Kakuzu vs Shikamaru and his team.
Naruto vs Pain (stopping before he goes to Nagato's lair)

Tokyo Ravens
Almost every battle involving Ohtomo-sensei.
Final Kon battle.

etc...

I should rewatch some of my favorites to validate my opinion on them, so I can eventually make a top X of some sort. If it weren't for school, I might have time, since there's not a lot of shows I'm excited about currently.
Seems like Autumn is going to change that though. All of those second seasons. I'm excited.

Off the top of my head, the things I tend to re-watch:

Kimi to Boku 2 ep 6 "Colorless Blue"
Baccano! Any episode involving Clair Stanfield/"Vino"/Rail Tracer/Felix Walken
Tokyo Raven last ep
SAO Sachi/Cilica/Lizbeth/Yui episodes + 1st dual wield battle
Sakamichi no Apollon (Kids on the Slope) School Festival + Last episode
Clannad up to end of Fuuko's arc.
Code Geass 2 ep 19?~24
Kara no Kyoukai Satsujin Kousatsu (Part 1)
Anohana last ep
Nodame Cantabile Finale ep 1
Beck ep 5, 10, 12, 23
Mahou Sensei Negima OADs (Ala Alba and Another World)

Most movies not included.
A lot of the other stuff I like I just don't bother with rewatching the anime and end up rereading the manga/novel.

While Chezlaw is my favorite character of the show, I do feel that Specter is the most entertaining one to watch. Claire is okay, and he grows into a rather unique and impressive character. However, I instantly fell in love with Graham when he showed his face to the screen.
Edit: Wow the Spanish dub of him is disappointing.

The series as a whole is just full of great characters.
Or, more generally, everything Narita Ryougo writes come with a good cast of characters.
It's a shame I seem to lose interest in reading them at ~vol 9 on all his series.
Haven't tried Vamp! yet though.

I'm looking for a show with a good plot that makes you think, ideally with minimal fridge logic, fanservice, and episodes where the plot ends at the exact same place it was when the episode started. Any ideas?

Katanagatari.

Not sure what you mean by "the plot ends at the exact same place it was when the episode started".
Example?

If I understood it correctly, then he wants the plot to advance with each episode. So, nothing like Kyoukai no Kanata's filler episode 6. And not Hunter x Hunter chimera arc where they spend several episodes on one minute.

Or maybe you could go with Durarara, since you liked Baccano.
Or Guilty Crown, which I feel is mandatory to watch, but I know that's far from true.

Watched episode 1 of katanagatari. I forgive you for HxH now.

wow, okay.
I didn't even like Katanagatari all that much.

Seriously? I always rated Katanagatari fairly highly.

I talked with cato about this already, but essentially, I have a high opinion of the anime, but I didn't derive much enjoyment from it. Sure, the last episode is one of the best episodes I have ever seen, but there wasn't anything about the show that catched me. It's more or less that I've come to accept that it has qualities that I don't necessarily care much about.

Went on one of my rare trips to Youtube and found a lot of nostalgic stuff.
They're mostly kids stuff, but man. I never thought I'd ever see Alice SOS ever again. The thing wasn't even a full show.

Katanagatari gets into a bit of a slump during the middle episodes.

Okay, so a bit more detail for my need of suggestions.
I'll watch just about anything, but I love love LOVE mecha anime more than literally anything.
And yes, I am in quite a rut.
Also, someone throw some sort of high school drama at me cuz I feel like watching some of that stuff.

I really want to recommend Valvrave, but I know that I shouldn't.
Do you like trainwrecks with high production value? If so, go with VVV.
Aldnoah.Zero was kind of cool.

White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2, White Album 2

I quite liked Kokoro Connect. Where "quite liked" means "It's my favorite show", but I don't think that applies to everyone.
You can also consider Golden Time and Nagi no Asukara I suppose.