With two moves, you can cover ten types with a ground move in freeze dry
When accounting for no dual-types (typing in general), the most efficient grouping of types I found was:
Freeze Dry: Water, Ground, Dragon, Flying, Grass
Ground: Electric, Rock, Steel, Poison, Fire
Fighting: Normal, Dark, Ice
Dark/Ghost: Ghost, Psychic
Steel: Fairy
Flying: Bug, Fighting
However, accounting for all known types you'll need Mold Breaker (to break Eelektross's levitate) and in addition to the types listed above
Fairy: (for Dark/Ghosts, Dark/Psychic and Kingdra)
Fire: For Bug/Steels and Steel/Flying
Rock: For fire/flying
If there's a more efficient combination or I missed something feel free to tell me I'm wrong
However, this was all a lie, because you really only need three moves to hit every Pokemon superffectively
- Soak
- An electric move
- A grass move