I think this is it:
A happiny with an attack lowering nature, 0 attack ivs, 0 attack EVs, soaked, -6 attack, max happiness using frustration against a mega aggron with +6 defense, full defense ivs, 252 defense EVs, and defense boosting nature.
Technically the lowest amount of damage possible is 0, if you use Hold Back or False Swipe on a Pokemon with 1 HP remaining.
If the target has 1 HP remaining, False Swipe will hit and leave the target at 1 HP. - Bulbapedia
There is also a bug in Gen 1, and a different one in Gen 5, that allows you to deal 0 damage in particular circumstances. However, these are clearly development oversights.
Excluding all of the above, the lowest amount of damage is 1. The number of ways to achieve 1 damage is practically infinite, so no need to list them here.
#1HP.
The answer is obvious and there are probably alot of matchups to achieve 1HP damage.