I was initially tempted to answer this one with AIC.

If you click on the link and scroll down you’ll find appallingly long list of good titles with terrible sequels, spin-offs, and reboots. AIC is a terrible example of franchise addiction at its worst.

But in the end I’m going to go with Gainax (whilst betting that rdmasters saw that coming a mile off).

Repeat after me: real women do not look, move, or act like that.

Also endless fanservice does not a story make.

The really annoying thing is that Gainax seems to have influenced a lot of other studios who now copy and/or exceed their worst excesses. (Repeat after me: real women...)

Speaking of Gainax’s worst excesses let us have a little look.

Gunbuster: works despite the fan service not because of it.

Nadia: Secret of Blue Water: a prime example of needing to edit (you can start by dropping the entire islands sequence, as also noted by seawasp here).

Neon Genesis Evangelion: Don’t go there. Just don’t. Watch RahXephon or Gasaraki instead.

He Is My Master: I haven’t gone there. Based on what I’m seeing at TV Tropes, there is no chance in hell that I will go there. And neither should you.  

And then there’s the title that Gainax is apparently working on now. So very tasteful. Not.

So there you have it: Gainax an anime studio that probably needs to be slapped good and hard.

Repeatedly.

Day 1 - Very first anime: Robotech, Gall Force, Vampire Princess Miyu.
Day 2 - Favourite TV series: Card Captor Sakura.
Day 3 - Your anime crush: Oh grow UP already.
Day 4 - Favourite Film: My Neighbour Totoro.
Day 5 - Anime character you feel you are most like (or wish you were): Kinoshita, Rocket Girls.
Day 6 - Most annoying anime character: Lynn Minmei, Super Dimensional Fortress Macross.
Day 7 - Character you’d most like to slap: Hiyama Hikaru, Kimagure Orange Road.
Day 8 - Most epic scene ever: Launch day - Wings of Honneamise.
Day 9 - Saddest anime scene: The passing of the Fortune Twins - Nariya and Eriya.
Day 10 - Favourite slice of life anime: ARIA.
Day 11 - Favourite mecha series: Patlabor: The Mobile Police.
Day 12 - Best instalment of a franchise you don’t like: Mobile Suit Gundam 0080: The War In The Pocket.
Day 13 - Worst instalment of a franchise you do like: WXIII: Patlabor the Movie 3.
Day 14 - Current (or most recent) anime wallpaper: I think I had a Silent Mobius wallpaper once.
Day 15 - A reality check for a genre you like: Romantic/Heroic Space Pirates.
Day 16 - An anime classic that simply doesn’t work for you: Ghost in the Shell (film).
Day 17 - Best Background Music: Silent Mobius 2: The Motion Picture.
Day 18 - Best Five-team: Silent Mobius.
Day 19 - Change you’d like to make to any one anime: Add an animation budget to Lodoss Wars.
Day 20 - Most ambiguous anime character: Reki, Haibane Renmei.
Day 21 – Favourite OAV Series: El Hazard the Magnificent World.
Day 22 - Animation studio you'd most like to slap: Gainax. 
Day 23 - Anime you think had the best, or most intriguing art
Day 24 - Favorite anime hero or heroine
Day 25 - Best anime villain
Day 26 - Your favorite harem anime
Day 27 - Favorite anime opening theme song
Day 28 - Favorite anime companion/summoned creature
Day 29 - Best Anime Music Video
Day 30 - Favorite anime ending theme