For this I will be counting white as a part of the Pokémon's colour palette, and I will be counting different shades of colour as different (e.g. red and dark red) so long as they are distinct from each other. I will not be counting shadows/highlights.. Black pupils will not count, and neither will tongues. So, without further ado:
Pokémon with 9 different colours:
Mega Steelix: chrome, gunmetal, dark grey, white, blue, black, pale yellow, pale blue, pink
Xerneas: deep blue, blue, black, white, pale yellow, canary yellow, orange, red, lavender
Toucannon: black, white, red, orange, orange-yellow, yellow, grey, dark blue, light blue
Comfey: green, orange, warm yellow, light grey, white, red, yellow, pink, blue
Magearna-Original: red, white, pink, light pink, blue, light blue, black, gold, yellow
Runners-up, with 8 colours:
Torterra: green, white, red, grey, light brown, dark brown, forest green, black
Silvally: grey, light grey, red, black, green, blue, purple, white (changes colour)
Necrozma: black, white, red, orange, blue, light blue, purple, green
Magearna: white, grey, yellow, pink, light pink, blue, light blue, black
Celesteela: jade, light jade, yellow-green, yellow, white, blue-grey, turquoise, dark green