How many countries have six or more colors on their national flag?

Two independent countries, although South Africa's is the most clearly discernible, with a black triangle at the pole side outlined in yellow, within a green horizontal Y shape outlined in white, with red above and blue below. The other, Dominica, has a small, partially blue parrot on a red medallion over a cross made of yellow, black, and white stripes on a green background.

See http://www.allstarflags.com/facts/worlds-ten-most-colorful-flags/ for their images, along with several other (mainly 5 colour) elaborate flags.

According to this web site, I count 14.

https://en.wikipedia.org/wiki/List_of_flags_by_number_of_colors

Good point, star_gazer, with the detailed coats of arms or other (usually small) symbols included in many flags, there are definitely more than two. Most of these would not be easily distinguished from any distance, but they are there. From the https://en.wikipedia.org/wiki/List_of_flags_by_number_of_colors site you provided, I count 2 qualifying national (not state or military) flags that are clear from a distance (Republic of South Africa and South Sudan) and 10 with two or three dominant colours supplemented by extra colours in small details in the symbols (Andorra, El Salvador, Croatia, Dominica, Haiti, Costa Rica, Equatorial Guinea, Fiji, Guatemala, and Belize).