I'd put this in order if I wasn't a lazy piece of shit.

I don't think you will ever beat Ai wo torimodose :)

Naruto had a lot of good openings. Shippuden openings 3, 5, 6, 7, and a few others were very memorable.

Evangelion is also a classic

Started with embeds, but ended up with way too many. Out of the ones I've watched, some of my favorites:

Cowboy Bebop Intro (over all #1)

Cowboy Bebop Outro

Cowboy Bebop Movie Intro

Cowboy Bebop Movie Outro

Fullmetal Alchemist Intro

Fullmetal Alchemist: Brotherhood Intro

Ghost in the Shell Intro

Ghost in the Shell: Standalone Complex Outro

Ghost in the Shell: SAC 2nd Gig Intro

Lupin The Third Intro

The Big O Outro (simple but classy)

Jojo's Bizarre Adventure Outro (just too catchy)

Samurai Champloo Intro

Moribito Intro

Code Geass Intro

Attack On Titan Intro

Ergo Proxy Intro

Hellsing Intro

Outlaw Star Intro

Baccano Intro

Semi-Related:

Avatar Book 3 Trailer

Batman: The Animated Series Intro

Doug Intro

Samurai Jack Intro

Simpsons 80's Couch Gag

Toonami - Broken Promise

Boomerang - Circles

Boondocks Intro

Urbance Trailer

Tales of Vesperia Intro

Cheers

https://youtu.be/qaLeO-2Fytg

Kill la Kill opening 2

only in alphabetical order (these swap places all the time in my mind):

Bleach - OP3 - Ichirin no Hana | https://www.youtube.com/watch?v=k3-h7CkgMSU
Break Blade - OP - Fate | https://www.youtube.com/watch?v=rpHBjFuTuuY
Busou Renkin - OP - Makka na Chikai | https://www.youtube.com/watch?v=PTjuAPFySdw
Gundam 00 - ED1 - Ash like Snow | https://www.youtube.com/watch?v=_d_for2jnQ0
Heroic Age - OP - Gravitation | https://www.youtube.com/watch?v=1ULGfQp_XSA
Iketeru Futari - OP - Fall in You | https://www.youtube.com/watch?v=cdYPJ0EQLXg
Kami-sama Dolls - OP - Fukanzen Nenshou | https://www.youtube.com/watch?v=_AA0icW3w2w
Slam Dunk - ED2 - Sekai ga Owaru made wa | https://www.youtube.com/watch?v=b77CA-1c6Gk
Terra e - OP2 - Jet Boy Jet Girl | https://www.youtube.com/watch?v=7XeWzK8ohC0#t=50s
Witch Hunter Robin - OP - Shell | https://www.youtube.com/watch?v=raSnXXBtCbc

meh, youtube compression makes some of these sound too flat for my liking

Duvet (serial Experiments Lain OP)

https://www.youtube.com/watch?v=t9CXmEUwvgM

Hanaji (Maria Holic OP)

https://www.youtube.com/watch?v=pOHwVZc6QYY

Ride on a Shooting Star (FLCL ED)

https://www.youtube.com/watch?v=xCKtq_hANrw

Changing (The Wallflower ED)

https://www.youtube.com/watch?v=Q9VEPCFzYmk

these work extremely well as 1:30 version along with

Elfen Lied - OP - Lilium

Gantz - OP - Super Shooter

Ghost in the Shell SAC - OP1 - Inner Universe

but unfortunately in my opinion these songs fall flat as full versions so they don't make my list as short versions always leave me unsatisfied especially if they are as awesome as these

Literally every OP/ED by MYTH&ROID

Overlord ED
https://www.youtube.com/watch?v=0Q0ta3tigPo

BBK/BRK ED
https://www.youtube.com/watch?v=1E4O5YFarYQ

Re:Zero ED
https://www.youtube.com/watch?v=1Y9PrCA86oE

Re:Zero OP2
https://www.youtube.com/watch?v=tUC578VmldU

Youjo Senki OP
https://www.youtube.com/watch?v=CMwlVyfy3m4