In regards to my mentioning Normal/Ghost, its only weakness is Dark. There were no Dark-types introduced until Generation 2. So far, this combo is the only one to have one weakness and hasn't been introduced yet.
As for pure Normal-type Pokémon, they are only weak to Fighting. The only way to lose this weakness is to use the move Trick-or-Treat on a pure Normal-type Pokémon, which will turn it into a Normal/Ghost-type Pokémon.
It's funny that, since there are no Normal/Ghost-type Pokémon introduced into the franchise as of yet, there is a way to turn a pure Normal-type Pokémon into a Normal/Ghost-type Pokémon.