Lain is a classic. Highly influential piece of pop culture. And even though its story is metaphorical it accurately predicted our incredibly online age, with its strange cultural and psychological problems.
The best part about the Nichijou OP is that Hyadain sang all the parts himself. He just pitched his voice up for the parts that sound like female backing vocals. Lends an extra touch of Nichijou-type humor to the whole thing.
I usually watch an OP once for a show and then skip it each time afterwards. That being said, there are a handful that I remember making the choice to always watch each episode. In no particular order:
I wrote a little bit about it a while back when I posted a clip for Thanksgiving. Overall, it is a piece of lighthearted fun that was perfect for watching an episode before bed. It also tried to be educational about how to properly work out, while at the same time using those educational segments as an excuse to include a bit of ecchi.
I didn't know or care about the sport at all going in, but the show presented it in such a way that you can't help get invested in the characters. On top of that it's just a excellently told story with a lot of good drama, and I think it says a lot about people and their relationship to sports/hobbies/art/etc. I hope other people will give it a try. :)
Really making me wrack my brain here, but that's what I got for now, tried to pick a few that a lot of folks probably wouldn't be all too familiar with. Of course there's plenty more I like that have already been mentioned here like Sorairo Days and the like.
Cruel Angel Thesis - Evangelion
Kuusou Mesorogiwi - Mirai Nikki
Bacchikoi amd Heroes Come Back - Naruto Shippuden
Basically Everything from JoJo's Bizarre Adventure. If I had to pick, sheet, I'd say Stone Ocean. Canzioni Preferite is the best non-opening song though!
Kaguya opening 1 (Love Dramatic) for it's smoothness
One Piece Opening 13 (One Day), it has a very "complete" feeling for how encompassing the arc is
Gundam Wing Opening 2 (Rhythm Emotion), is basically what made me enjoy music as a kid
Full Metal Alchemist (OG) opening 2 (Ready Steady Go) . It's got so much energy
Yu-gi-oh opening zero (Kawaita Sakebi). For some reason it's become so iconic that I think it's more well known than the openings for the mainline series (I blame abridged but I remember listening to it before abridged came out)
Going through this exercise, I really like jazzy, hip hop, and, sometimes, heavy metal openings. (Still get chills from Attack on Titan's "Rumbling" OP.) I've hyperlinked each song below for specificity (and nostalgia).