Most Anime have started at first as a Manga, yet there are some exceptions, which did not start off like that!
Basically I'm counting everything that didn't originate from a Manga or Light Novel! It's fine, if it originated from something else and also if a Manga adaption was later created!
1 Cowboy Bebop
Cowboy Bebop is a 1998 anime television series animated by Sunrise featuring a production team led by director Shinichirō Watanabe, screenwriter Keiko Nobumoto, character designer Toshihiro Kawamoto, mechanical designer Kimitoshi Yamane, and composer Yoko Kanno.
Most of the anime on this list actually do have a manga series
2 Angel Beats!
Angel Beats! is a 13-episode Japanese anime television series produced by P.A.Works and Aniplex and directed by Seiji Kishi.
3 Neon Genesis Evangelion
Neon Genesis Evangelion, commonly referred to as Evangelion or Eva, is a Japanese animated television series produced by Gainax and Tatsunoko Production, and directed by Hideaki Anno.
4 Gurren Laggan
Most over the top anime I've ever seen.
Hmm... isn't it spelled Gurren Lagann? Maybe it's spelled differently in other countries? - ModernSpongeBobSucks
JUST WHO THE HELL DO YOU THINK I AM?! - ModernSpongeBobSucks
5 Code Geass
Code Geass: Lelouch of the Rebellion, often referred to as simply Code Geass, is a Japanese anime series created by Sunrise, directed by Gorō Taniguchi, and written by Ichirō Ōkouchi, with original character designs by manga authors Clamp.
6 Kill La Kill
Kill la Kill is an anime television series produced by Trigger. In Kill la Kill, Ryuko Matoi transfers to a new high school in order to find her father's murderer.
Don't lose your way with this over-the-top turn on the magical girl genre! - ModernSpongeBobSucks
Uh...kill la kill DID origonate from a manga...
Best magical girl anime since Madoka Magica and Sailor Moon. And a motherlode of fanservice as well! - ModernSpongeBobSucks
Steins;Gate is a 2011 anime television series created by the animation studio White Fox based on 5pb. and Nitroplus's 2009 video game of the same name, and is part of the Science Adventure franchise.
8 Puella Magi Madoka Magica
Puella Magi Madoka Magica is a 2011 Japanese anime television series produced by Shaft and Aniplex, directed by Akiyuki Shinbo, written by Gen Urobuchi, music by Yuki Kajiura, opening sang by ClariS, ending by Kalafina (sometimes by the seiyuus) and original characters by Ume Aoki. It also has 3 movies, ...read more.
Definitely this show. It deserves to be number one. If you gave me a week I could maybe say everything great about this show, but I don't wanna be here forever. Madoka Magica is a masterpiece, and I will forever love this amazing anime
Words cannot describe how great this show is
This has been my personal favourite anime for almost five years now, and for good reason. definitely worth a watch. it's only twelve episodes, but boy, is it an emotional roller coaster. you might not want to watch this one too quickly unless you want to spend the next hour after completing the series staring at a wall with empty eyes.
9 Samurai Champloo
Samurai Champloo is a Japanese anime series developed by Manglobe. Samurai Champloo is set in an alternate version of Edo-era Japan with an anachronistic, predominantly hip hop, setting. It follows Mugen, an impudent and freedom-loving vagrant swordsman; Jin, a composed and stoic rōnin; and Fuu, a ...read more.
Haven't seen it yet. But it's airing on Toonami as of January 2016. I heard the anime influenced The Boondocks. - ModernSpongeBobSucks
Psycho-Pass is an anime television series that was produced by Production I.G, directed by Naoyoshi Shiotani & Katsuyuki Motohiro and written by Gen Urobuchi.
? Magical Girl Lyrical Nanoha
Fun Fact: This was actually spun-off from an eroge (porn) game called Triangle Heart 3 in which the titular heroine was introduced as a minor character. - PerfectImpulseX
11 Darker Than Black
Darker than Black, known in Japan as Darker than Black: The Black Contractor, is an anime television series created, directed and written by Tensai Okamura and animated by Bones. In 2007, the 26 episode anime named Darker Than Black: The Black Contractor aired with a huge commercial success. It was ...read more.
Pokémon, abbreviated from the Japanese title of Pocket Monsters and currently advertised in English as Pokémon: The Series, is a Japanese anime television series, which has been adapted for the international television markets.
13 Space Dandy
14 Serial Experiments Lain
Serial Experiments Lain is an avant-garde anime series directed by Ryutaro Nakamura, with character design by Yoshitoshi ABe, screenplay written by Chiaki J. Konaka, and produced by Yasuyuki Ueda for Triangle Staff. It was broadcast on TV Tokyo from July to September 1998. The series is influenced by ...read more.
A very cool Psychological Anime I started watching recently! Extremly intense and deep storyline! Overall a anime worth watching! - CastlevaniaFanboy128
FLCL is an anime original video animation series written by Yōji Enokido, directed by Kazuya Tsurumaki and produced by the FLCL Production Committee, which consisted of Gainax, Production I.G, and King Records.
Bleach is a Japanese anime series based on Tite Kubo's manga of the same name. Bleach adapts the first 54 volumes of Tite Kubo's manga series of the same name. In addition, 4 anime-original arcs were broadcast in-between and during the adaptations of original arcs of the manga. ...read more.
Uh actually there was a bleach manga to be exact 74 of them
19 Little Witch Academia
20 God Eater
God Eater is an anime adaptation of the God Eater video game series. It is animated by Ufotable and began airing on July 12, 2015 after a one-week delay.
This actually originated from a PSP game franchise. - SelfDestruct
By the way this show is amazing
21 Guilty Crown
Guilty Crown is a 2011 Japanese anime television series produced by Production I.G which aired on Fuji TV's noitamina program block from October 2011. The story revolves around Shu Ouma, a high school boy who inadvertently obtains an ability called "Power of the King" that enables him to draw out items ...read more.
22 Panty and Stocking with Garterbelt
Blood+ pronounced "Blood Plus," is an anime series produced by Production I.G and Aniplex and directed by Junichi Fujisaku.
My favorite anime of all time.
Best female lead (Saya), best sidekick (Hagi), best backstory, best action, best weapons.
This anime is just amazing.. and outstanding.
This anime places such a warm feeling in your chest
25 Yuri on Ice
Yuri!!! on Ice is a sports anime television series about figure skating. The anime was produced by MAPPA, directed by Sayo Yamamoto and written by Mitsurō Kubo.
26 Wolfs Rain
“Digimon” is a term commonly used to describe a variety of Japanese anime series based on the Bandai virtual pet brand “Digimon” (Dejimon). As of 2017, there are 7 series: “Digimon Adventure” (1999), “Digimon 02” (a.k.a. “Digimon Adventure 02”) (2000), “Digimon Tamers” (2001), ...read more.
28 Rage of Bahamut: Genesis
I am sure a lot of people haven't heard of it but it originated from an iOS game with the same name (warning: English-language version of Rage of Bahamut was shut down). - SelfDestruct
29 Devil May Cry
30 Death Parade
Death Parade is a 2015 Japanese anime television series created, written, and directed by Yuzuru Tachikawa and produced by Madhouse.
Danganronpa: The Animation is a 2013 anime television series based on Spike Chunsoft's murder mystery video game, Danganronpa: Trigger Happy Havoc.
Free! is a Japanese anime television series directed by Hiroko Utsumi and produced by Kyoto Animation and Animation Do.
34 Kemono Friends
35 Terror in Resonance
36 Show by Rock!!
37 Uta no Prince-sama
38 Sakura Quest
41 Celestial Method
42 Boku No Pico
Boku no Pico is an anime series of three OVA episodes created by Katsuhiko Takayama. It was released to DVD on September 17, 2006.
43 Kado: The Right Answer
44 Ground Control to Psychoelectric Girl
45 Your Name