It's kind of hard for me to articulate my love for the Rolling Stones. When I was six years old, I told my parents that I wanted to be Mrs. Jagger-Richards when I grew up. It was admittedly kind of a whoreish goal to have in life -- to marry two of the Stones frontmen (preferably at the same time), but it seemed like a dream in my eyes.
My favorite period of the Rolling Stones is when Brian Jones was in the band, which was up until 1969. Yes, musically and artistically the band may had grown more during the seventies and eighties, but I still believe they were at their best in the sixties. They had this great rock vibe about them -- like Mod-meets-street-meets-R&B-meets-Psychedelic rockstar... if that makes any sense.
I also believe that the boys had the most fun during the sixties. Not like drug-related fun (though they had their fair share of that during the 60s too), but fun like they were really actually enjoying each other's company and the music. I don't know, maybe that's just me though.