I had to think long and hard about what to do for Rainbow. I decided it would be an excellent opportunity (like this whole challenge, really) to use several of my untried polishes. Here's what I finally came up with:
For the sake of because I felt like it, I decided to use Wet & Wild polishes for as much of this as possible, which means everything except orange. But then I decided to use an untried white for the base instead of French White Creme (which, turns out, is a better/less streaky formula... oh well now I know), I used L.A. Colors - Energy Source.
Going from red -> purple, the polishes used (all Wet & Wild except for orange, as noted):
Sinful Colors - Patrika
The Wonder Yellows
Sage in the City
Jezebel, Blue Bijou, and Eggplant Frost are from the Wild Shine line, while The Wonder Yellows and Sage in the City are Fastdry.
After doing a sideways gradient to try to blend the colors together across all the nails, which turned out a little rougher than I would have liked, I decided it needed more bling so I added a coat of Wet & Wild Wild Shine - Hallucinate, an iridescent-holo glitter topcoat.