I I don't think there is a combo that has no weakness. I used the poketech app that lets you see what types are effective or not against other types in my pearl game and found none with only one super effective or no super effective weaknesses.

Normal/Ghost has (based on the poktech app on pearl):

Normal, fighting, ghost - not effective (resistant)

Fire, water, electric, grass, ice, ground, flying, psyscic, rock, dragon, steel - regulary effective

Poison, bug - not very effective

Dark - Super effective

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Ghost/Dark has (based on the poktech app on pearl):

Normal, Fighting, Psychic - not effective (resistant)

Fire, water, electric, Grass, Ice, Ground, Flying, Bug, Rock, Ghost, Dragon, Dark, Steel - regulary effective

Poison - not very effective

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I'll suggest Electric/Ground. Neutrals out Ground's weaknesses to Grass and Water, as well as Electric's own weakness to Ground. It is also able to hit Flying-types, which are usually immune to Ground.

I also checked out a ground/Ice and found Fire, Water, Fighting, Grass are all super effective against it. Now electric/ground has water, Ice, Ground and grass which are super effective against it. SO Gras is still, along with water, super effective against it. But it can hit flying types, with electric moves which are super effective on most flying types except zapdos.

So as far as I can see, there is not type combos that have no weaknesses, if other combos were tried you'd mostly get a wider move pool for that pokemon that could hit super effective moves on pokemon it before could has no super effective types that can be used on it and I am not sure if thats the only one.