Question

Could there be a moveset that could cover all types?

Comments

Excluding Smeargle.

Answer 1

No because there is no moveset that can do supereffective damage to Sableye or Spiritomb unless you run Foresight or Odor Sleuth (and why would you xD???)

Some types weakness/resistance. ance each other so its always possible that you won't have anything for a certain type combination.

Comments

you forgot eelektross :)
ability levitate

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Mold Breaker deals with that.

Answer 2

No, except if the moveset contain Metronom or Judgement.
Source: Experience

Comments

And the bast type coverage I can find is Ground/Fighting/Ice/Electric.  It hits like 520+ super effectively and just over a hundred normal effectively. The draw back is it cannot hit shedinja :L
But the set of a Grass/Fighting/Rock/Flying hits a few less super effectively and can hit shedinja

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The best type coverage I have found was Ground/Fighting/Electric/Flying (126 normal effectively and 543 I think).

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Ground/Ice/Electric/Dragon.... Zekrom, Dragonite, Garchomp, or Kyurem-B all pull this off well.

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one of the four best one is ground/ice/fighting/flying, covers 14 of the 17  types, so something like + or - 550 super effective and the rest normal