Some good responses here. After listening to them all a couple times while playing Nethack I'd have to say these are my top picks from this thread.
[Xam'd Lost Memories Opening]
This one is very funky and I like it a lot without even having seen the anime.
TV Asahi held a poll in which almost 1800 foreigners voted for the best anime song, with Evangelion's opening coming in first.
www.siliconera.com
I haven't seen most of the anime listed but I was going to disagree with Attack on Titan getting the bottom 3 slots because I only remembered really liking the first theme, but after listening to them again actually I think those 3 are all pretty good. My dislike of the writing must have impacted my appreciation of the opening themes.
Of course, Cowboy Bebop is a classic and I'd be very surprised if someone hadn't mentioned it. I have it on my iPod so I can't disagree.
Big O. This song sounds a lot like Flash by Queen.
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Onegai Sensei.
Ghost in the Shell: Stand Alone Complex (both series)
The Onegai Sensei theme isn't outstandingly unique, I'm sure I've heard other themes that are fairly similar, but it's well done and I like it.
I like the first Ghost in the Shell theme you posted too. Reminds me a bit of the music from Nier: Automata.
I haven't seen either of these anime, though I should probably get around to watching Ghost in the Shell one day.
Of course the first open that FMA: Brotherhood used really nailed it (they should never have changed it)
And to round things out with a joke, that's also a good answer... each Toaru series did particularly well when coming up with an open. But (heh) one stands out above all others.
I of course agree with FMA: Brotherhood, as I think I've already done in the past.
As to To Aru Majutsu No Index II. Not sure why this is a joke answer, but this actually gets me a little interested in watching the anime because I've seen it around here and there. I know nothing about what it's about, though, is it worth watching?