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Suppose I have two vials of Basic Poison:

You can use the poison in this vial to coat one slashing or piercing weapon or up to three pieces of ammunition. Applying the poison takes an action. A creature hit by the poisoned weapon or ammunition must make a DC 10 Constitution saving throw or take 1d4 poison damage. Once applied, the poison retains potency for 1 minute before drying.

I take an action to apply one dose to my sword, and then another action to apply a second dose. Then I hit a monster with my sword. Does the effect accumulate, with each poison causing an additional 1d4 damage (for a total of 2d4 on top of the normal weapon's damage)?

There are rules about not being able to accumulate multiple of the same condition, or the same magical effect, but I'm not sure there are any rules about combining mundane effects like this.

If there's no official answer, I'd appreciate any reasoning to help me make a ruling as a DM. I have a player whose character recently acquired an Alchemy Jug (which can make Basic Poison), and has asked me how multiple doses at once would work. I'm not sure whether poison ought to be something that is either there or not (and multiple doses wouldn't change anything other than maybe extending the duration) or whether applying more poison ought to be more harmful than applying less. Both interpretations seem reasonable to me.

Game effects, not just magical effects, of the same name don't stack. See the DMG errata:

Combining Game effects (p.252) This is a new subsection at
the end of the “Combat” section:

Different game features can affect a target at the same time. But when two or more game features have the same name, only the effects of one of them—the most potent one—apply while the durations of the effects overlap...

So only one dose (1d4 damage) of the poison would apply.

The closest you can come to this is perhaps a double-bladed weapon and Extra Attack

As correctly pointed out by Purple Monkey this won't work by coating, e.g., the same sword more than once with basic poison and then attacking with it, if such poison counts as a "game effect" mentioned in the DMG errata (which it likely should be).

However, does the player have a double-bladed weapon, and Extra Attack? Then she could coat both blades right before jumping into the fray, and potentially get the 1d4 poison on both attacks (if they hit), because they no longer happen "at the same time."

Note: There are double-blade weapons in 5e now. Wayfinder's Guide to Eberron from WotC introduces the double-bladed scimitar. You don't even need Extra Attack to hit with the opposing end-blade (you can use a bonus action).

If you want to house-rule to allow double-dosing the poison...

...or if you read the DMG errata as being unclear on whether mundane poison is a "game effect", then as 16th century toxicologist Paracelsus said, "dosage makes the poison". In real life, greater dosage of poison usually means more damage, but it is a non-linear relationship between the two. And how much could stick to one's sword without just dripping off? It would seem hard to manage for the DM.