I'm pretty sure that the Pokémon with the least weaknesses and most immunities is Spiritomb and Sableye. They are Dark/Ghost types, with 3 immunities (Psychic, Fighting, and Normal), their only weakness is Fairy, and everything else does either not very effective damage or neutral damage.
EDIT: Drifloon/Drifblim also has 3 immunities (Fighting, Normal, Ground). Honedge/Doublade/Aegislash has 3 immunities (Fighting, Normal, Poison), Mimikyu has 3 immunities (Dragon, Fighting, Normal), and Golett/Golurk has 3 immunities as well (Electric, Fighting, Normal).
Source: knowledge + A Geeky Jawa's comment