dnd 5th – Can I apply several doses of Basic Poison to a weapon and get a cumulative effect?

The closest you can get is maybe a double-bladed weapon and Extra Attack

Suppose your goal is to get 2d4 instead of 1d4 of this poison damage in one turn, with one weapon and without applying the poison once the fight started.

As Purple Monkey rightly points out, this will not work by covering, for example, the same sword more than once with basic poison and then attack with, if such a poison counts as a "game effect" mentioned in the DMG errata (which should be the case).

However, does the player have a double-blade weapon and Extra Attack? Then she could smear the two blades just before jumping into the fray and potentially get the 1d4 poison on both attacks (if they hit), as they no longer occur "at the same time".

Note: The are double-blade weapons in 5th now. WotC's Guidefinder for Eberron presents the double-blade scimitar. You do not even need Extra Attack to hit with the opponent's endgame (you can use a bonus action).

If you want to allow double dosing of poison …

… or if you read DMG's errata as being unclear on the question of whether ordinary poison is a "gaming effect", then as the 16th century toxicologist Paracelsus said , "the dosage makes the poison". In real life, a higher dose of poison usually means more damage, but it is a non-linear relationship between the two. And how much could stick to his sword without draining? This would seem difficult to manage for the SM.