So, I just put together this little chart showing all the type combinations we have as of this generation.
To represent each combination I chose the first fully-evolved, non-legendary Pokémon of that type to appear in the National Dex.

*Combinations that only have been represented by legendary Pokémon or forms/Mega Evolutions are semi-transparent, as I don't feel like they really "count".

*The lines represent primary types, while the columns represent secondary types. Combinations that exist only in reverse order are indicated by a little "refresh" symbol. I know most people don't care about the order, and it doesn't change anything gameplay-wise, but I feel like it matter for the "flavour" of the Pokémon.

It's funny how Flying has been used as a secondary type for every type, yet only used as primary/mono twice, one of those being a legendary.

Some combinations that I'm dying to see are:

Grass/Fire
Fairy/Dark

It baffles me that those haven't been used yet: Bug/Psychic and Bug/Dark