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

The closest you can get to this is perhaps a double-edged sword and a multiple attack.

Suppose your goal is to get 2d4 instead of just 1d4 of that poison damage in a single turn, with only one weapon, and without reapplying the poison once the combat has begun.

As Purple Monkey correctly pointed out, this will not work when coating, for example, the same sword more than once with basic poison and then attack with him, if such poison counts as a "game effect" mentioned in the DMG errata (which probably should be).

However, does the player have a double-edged sword and a multiple attack? Then he could cover both blades just before jumping into the fray, and potentially get the 1d4 poison in both attacks (if they hit), because they no longer happen "at the same time".

If you want to dominate the house (or if you read the DMG errata as unclear about whether mundane poison is a "game effect"), as the sixteenth-century toxicologist Paracelsus said, "the dose makes the poison". However, there is a non-linear relationship between the dose of the poison and the effects (as was also observed in Paracelsus). Which means that the damage does not necessarily double when the dose is.