To answer you title's question (which is very different than what you're asking):
In the Basic Rules, under Hit points:
A creature's current hit points (usually just called hit points) can be any number from the creature's hit point maximum down to 0. This number changes frequently as a creature takes damage or receives healing.
There is no counting beyond 0 unless the DM home brews some rules.
To answer the Clay Golem example, here is the relevant text:
If the target is a creature, it must succeed on a DC 15 Constitution saving throw or have its hit point maximum reduced by an amount equal to the damage taken. The target dies if this attack reduces its hit point maximum to 0. The reduction lasts until removed by the greater restoration spell or other magic.
So Charlie starts with 15 hit points maximum, and the friendly cleric casts aid on them. Their max hit points for the duration is 20. At the end of those 8 hours, Charlie's hit points maximum goes down by 5 returning them to 15 hit points total, 15 hit points maximum.
The next day, Charlie is aid'ed again. But during those 8 hours, Charlie is slammed by a Clay Golem for 18 points of damage. Charlie fails the saving throw so their hit point maximum is now effectively 2 until a greater restoration spell is cast on Charlie.
Once again, at the end of 8 hours, the maximum hit points loses those 5 extra points. Without a greater restoration, that hit point maximum would be reduced to -3, but, as noted above, the game stops counting at 0.
Once aid ends, Charlie dies with 0 hit points and a maximum of 0 hit points.
And here is the first point of DM interpretation. Was it the slam attack that brought Charlie to 0 hp, or was it the fact that aid wore off? The distinction is important.
- If it is ruled that the slam attack reduced the maximum to 0, per the rules of the slam attack, Charlie would be instantly killed. No Death Saves, just dead.
- If it is ruled that the lack of aid reduced Charlie to 0, then Charlie is merely unconscious. They get Death Saves as normal.
As a DM, I'd rule that it would be equivalent to stepping out of an anti-magic zone. While aid is going, the slam effect is at bay, but once you lose that protection, the slam effect still exists and therefore the character is subject to its rules. For me, the key factor is that once aid is gone, the after effect of the slam is still in affect.
Also note that either way, Charlie has a 0 hit point maximum until greater restoration is cast. So even if Charlie makes Death Saves, they cannot gain a single point.
The next DM interpretation is, what if Charlie got the greater restoration spell, but wasn't healed? In that case, the hit point maximum would go back up to 20 (since aid is still in effect), but they would have 2 hit points. At the end of aid their maximum would go down to 15.
But would they lose those last 2 hit points?
Per this question, it is accepted that the character would lose the 5 hit points knocking Charlie unconscious.
As a DM, I have an alternate theory in that those extra hit points are used first, not saved until last. But that is not really part of the question so I'll leave that be.