Some favourites from things I've actually seen are below.
Jun, the opening from Cromartie High School:
===WARNING: IT'S NOTHING BUT SQUEAKY BABY VOICES FOR THE REST OF THIS LIST===
Either of the openings to Mitsudomoe, the most criminally underrated anime on the planet:
Aimai Net Darling, Akira Kogami's character song from Lucky Star:
I doubt anyone will listen to all these, so my last one will be Kyou Mo Heiwa Desu (It's Another Peaceful Day), another character song by the characters of Nichijou:
Okay, fine. A couple more from Nichijou. Firstly, one of Hakase's character songs:
And Zzz, the adorable lullaby from the end credits, as a slight palate-cleanser: