There are many great anime openings. There are (not as) many great anime endings. However, when an anime has both an excellent OP and ED that function together as a unit, it is something truly special. The OP serves the purpose of setting the mood of the show, usually an upbeat song that gets the adrenaline moving. The ED should serve the opposite purpose, not be distracting but act as a mellow sign off. There are a couple of exceptions on the list, but always there is a contrast between the two bookends that allow for a rounded feeling to the show.
I will refrain from making unnecessary comments. I simply post the songs and allow you to decide for yourself. Note that these were selected primarily on a musical basis, but the visuals did have some impact on my decision.
There are two honorable mentions that just missed the cut.
Honorable mention: Aria (Season 1):
Why it missed the cut: I love the OP/ED, but they're both the same style and serve the same function, so they're not really a "tandem" by the definition I set.
Honorable Mention: Magical Girl Lyrical Nanoha A's
Why it missed the cut: I had it on the list originally, but then was like "oh crap, I forgot about [insert anime here]."
10. Welcome to the NHK! (first set)
Puzzle/Odoru Akachan Ningen
9. Tsubasa Reservior Chronicle Season 1
8. Bakemonogatari (all sets)
I chose this one mostly because I loved all the OPs of this anime, and the ED was just amazing. For this list, I chose my favorite OP.
Renai Circulation/Kimi no Shiranai Monogatari
7. K-On!! (Season 2, second set)
Utauyo!! MIRACLE/No THANK You
Sorry for the flipped songs. They were the only TV size versions I could find.
6. Clannad (season 1)
After Story is my favorite anime of all time, but this tandem just edges out the After Story one.
5. Puella Magi Madoka Magica
4. Neon Genesis Evangelion
Cruel Angel's Thesis/Fly Me To the Moon
This was the one I forgot about (HOW?). I didn't really love the ED as much as the ones below, thus the lower rank.
3. Angel Beats!
My Soul Your Beats/Brave Song
2. Outlaw Star (first set)
Through the Night/Seihou Bukyou
As of this post, I just started watching Outlaw Star with my university's anime club. So far I haven't seen the other sets for the OP/ED, but this one is pure win.
And of course...#1 is
1. Cowboy BeBop
Tank/The Real Folk Blues
Here's a playlist of all the videos. Feel free to comment on any anime I idiotically omitted.