The potion of poison calls for a DC 13 Constitution saving throw. To automatically fail this saving throw, you have to have a modifier of -8. But I'm not sure a modifier this low is actually possible.
So, what is the lowest possible Constitution saving throw modifier?
I would like to know both permanent and temporary modifiers. Additionally, if using an effect such as bane, it only counts for the minimum possible penalty. Since bane substracts 1d4 from an affected creature's saving throws, that would only count for only a -1 penalty. The reason for this is that what I'm really looking for is to determine if it is possible to guarantee failure on the saving throw(s) against the Potion of Poison. If your modifier is -6, bane does not guarantee failure, as you could roll a 1, resulting in a modifier of -7. However, if your modifier is -7 before applying bane, bane then guarantees automatic failure.
Finally, the one drinking the poison is a player character, so the character build must follow the standard rules for character creation. We can assume any other dice rolls fall in favor of your setup (yes, you may assume you rolled four 1s when generating your CON stat).
Best Answer
-27 with some curse stacking
We are a character starting with a constitution score of 3. We're a Fey, such as by being a satyr (other fey races are available). Earlier today we were raised from the dead, in a temple of the gods we're still currently in, and since then we have reattuned to our Stone of Ill luck, wheel of stars, and some artefact with the major detrimental property:
(That ability score is of course Constitution)
We have also drawn the Euryale from the Deck of Many Things.
We then enlist a little help from some co-conspirators: A mite uses its Blood Boiling Hex on us, a monstrous peryton uses its Warp Shadow, a 6th level Wild Magic Sorcerer casts mind sliver, a 3rd level Eloquence Bard casts bane on us, and then uses their Unsettling words on us. The details don't really matter, each has us subtract a die from our saving throws (assuming we fail any previous checks. We also have a 7th level Artificer who's been the subject of feeblemind and a 6th level Druid of the Stars who's rolled woe for their Cosmic omen for the day. We then drink the poison to make a saving throw, and the sorcerer uses their Bend Luck, the Artificer their Flash of Genius, the druid their woe, and we use the Portent of the Stars (from the wheel of stars) ability to subtract a die from the saving throw.
Now, we have a constitution score of 1 for a base modifier of -5, Flash of Genius adds a -5, and raise dead applies a -4, and Euryale and stone of ill luck each a -2. We then have nine dice being subtracted, each counting for -1. This leaves us with a glorious guaranteed modifier of -27, making all saving throws DCs quiver with our aptitude towards failure.
(Footnote: We just need to make sure we have enough hitpoints to survive the mind sliver. Normally, 2d6 damage isn't such a huge threat, but it becomes something of a risk here. That might well be why we have to be raised from the dead.)
With a sufficiently low wisdom score, we'd never succeed the save against Warp Shadow (DC 14), which means we have a perpetual -10 (-5 -2 -2 -1d4) guaranteed constitution save modifier. If we never leave the temple of the gods and it's been made permanent we can push that to -11.