Fighting and Normal with Scrappy hit everything neutrally except Shedinja. Dragon and Fire with Turboblaze (to ignore flash fire) hits pretty much everything neutrally. The best combination actually comes from Feraligatr. As quoted from my post for his moveset: 'A lovely little moveset for maximum type coverage. Actually, there are 592 Pokemon that this guy can hit super-effectively against, he only has two weaknesses himself and he has fair speed and good bulk meaning he's a very versatile player. Anyway, my type-advantage Feraligatr.
[email protected] Belt/Life Orb
Ability: Sheer Force
Brick Break/Dragon Dance
Explanation: Basically, this guy hits every type (based on the moves on the left) apart from Water, Fighting, Bug and Fairy (and a few dual types resist him, such as Charizard and Malamar), but every single other Pokemon (including those dual-types) is hit for neutral damage. Expert Belt takes advantage of this, it boosts the power of super-effective moves by 20%, which is basically giving Feraligatr a Life Orb with no recoil. That brings us to the Life Orb, thanks to Sheer Force Feraligatr gets no recoil on Sheer Force boosted moves (Ice Punch/Fang, Crunch and Bulldoze), which brings us to his moves. Shadow Claw gives you coverage against about 12 more Pokemon, but doesn't abuse the Sheer Force/Life Orb combo, whereas Crunch does. Ice Punch is more accurate, but Ice Fang has more Power (both abuse the SF/LO combo). Brick Break is coverage and gets rid of Reflect from Klefki etc. but isn't affected by Sheer Force. Earthquake has higher base power, but isn't boosted by Sheer Force so it will have recoil, whereas SF boosts Bulldozes power to 90 which is 10 less, but the LO won't give you any recoil. Your choice on the moves. This moveset makes him a lethal opponent. If you want, you can swap Brick Break for Dragon Dance and use his bulk to set up, but this will reduce the number of opponents you can super-effectively hit by just over 100 so is not recommended.'
Edit: Some other good combos:
Dark/Electric - Hits all Pokemon neutrally (489) or super effectively (274) apart from 15 Dual-Types. Used by Absol (Night Slash or Dark Pulse/Thunderbolt), Lucario (Dark Pulse/Thunderbolt) etc.
Ice/Ground - Hits 410 Pokemon super effectively, and the rest standard apart from Surskit and Shedinja. Stab by Mamoswine (Earthquake/Ice Shard or Blizzard etc.)
Fairy/Dark - Hits 221 super effective, 554 standard and 3 not very effective (Mawile, Klefki and Mega-Mawile). Used by Gallade (Night Slash/Draining Kiss)
Ground/Flying - Hits 372 Super, 396 standard and 10 not very effective. Stab by Gliscor, used by Dragonite, Salamence etc. (Earthquake or Bulldoze/Aerial Ace or Fly or Acrobatics)
Flying/Ghost - Hits 284 Super, 488 normal, 6 not very effective. Used by Sigilyph (Air Cutter/Shadow Ball), Banette (Shadow Claw/Aerial Ace), Absol (Shadow Claw/Aerial Ace) etc.
There's 8 for ya, including Feraligatr. I was aiming for the lowest amount of not very effectives, there are type combos that have more super effective hits, however they also have more not very effective hits.