Mainstream anime was a staple of my childhood so the ones that immediately came to my mind are pretty unoriginal.
Of course, you have Dragon Ball Z's "Chara Hetchara", an absolute classic. Evangelion's opening "A Cruel Angel's Thesis" also doesn't need an introduction. Other equally good ones imo are Yu Yu Hakusho's "Smile Bomb" and Rurouni Kenshin's "Sobakasu." I thought Sobakasu was so good I even got into the band that sung it, Judy and Mary. As a total Pokemon kid, the original Japanese opening theme "Mezase Pokemon Masuta" remains dear to me. Last but not least, Two Mix's Gundam Wing themes "Just Communication" and "Rhythm Emotion." are also top notch.
Speaking of Two Mix, my all top favorite show for theme songs and one of my favorite anime in general is Detective Conan. Two Mix had a fantastic track called "Truth - A Great Detective of Love" as the show's 5th opening theme. A fun fact is the singer of Two Mix, Minami Takayama does the voice of the titular character as well as many other super famous anime characters.
Anyway, Detective Conan's had some of my favorite opening themes including what I'll say for now as my all time favorite. It's call "Mysterious Eyes" by Garnet Crow and it was the show's 7th theme song.