Gainax

Background
Gainax, Inc. was founded on December 24, 1984 as "Daicon Film", eventually changing their name to "Gainax", basing the name after the Tottori Prefecture term for "giant" ("gaina") with an "x" added to it, as this sounded "good and international". Notable anime produced by this company includes Gunbuster, FLCL, Tengen Toppa Gurren Lagann, Panty and Stocking with Garterbelt, and, most notably, Neon Genesis Evangelion, which have all gained critical acclaim and have been commercially successful. Unfortunately, the company is currently a shell of its former self--after Panty and Stocking was put on hiatus, many Gainax alumni left the studio to make their own companies, including Trigger (which features many key Gainax animators). The company has since refocused itself on producing lesser-known anime and manga.

1st Logo (1987)
TBA. Seen on Royal Space Force: The Wings of Honneamise.

2nd Logo (19??)
TBA. This logo is possibly a myth, though it does look legitimate. Possibly seen on Gunbuster or another early Gainax work, or at least a DVD release of it.

3rd Logo (July 19, 1997)
Logo: On a black background, the grungy white text "GAINAX" appears at the bottom-right corner of the screen and vibrates. It appears for a split-second.

FX/SFX: The text briefly appearing and vibrating.

Music/Sounds: A record player sound effect.

Availability: Seen only on The End of Evangalion.

Editor's Note: This logo is very hard to see due to the vibrating text and it flashing, not to mention the logo's very brief runtime. That said, knowing the teasing nature of Gainax alumni, and especially the extremely dark nature of the movie this precedes, this was all probably intentional.

4th Logo (2000)
Logo: Scrolling up in the credits underneath the Production I.G. logo, we see the text "GAiNAX" stylized much like in the 2nd logo but in a thinner font.

FX/SFX: It's an in-credit logo.

Music/Sounds: The closing theme of the show ("Ride on Shooting Star" by The Pillows).

Availability: Seen on FLCL.

5th Logo (199?- )
Logo: Just an in-credit text that reads:

PRODUCED BY GAINAX

It can also have "ANIMATION PRODUCED BY" above the company name.

FX/SFX: Same as the previous logo.

Music/Sounds: Same as the previous logo.

Availability: Seen on Panty and Stocking with Garterbelt, among others.