For the sake of an example, assume I'm a level 1 Sorcerer or Wizard who's somehow had the misfortune of winding up with 1 CON. Maximum HP at level 1 is determined by the highest possible roll on your hit die plus your CON modifier. For me, that's \$6+(-5)=1\$. Horribly low, but technically survivable. I just have to not get hit.
The problem arises when I level up. For simplicity, let's say my group doesn't roll hit dice to determine how HP changes. We instead use the alternate option of adding the listed flat values from the class descriptions in the Player's Handbook (the average roll on your hit die, rounded up). Using this method, the formula for what's added to my maximum HP is: $$\left (\frac 6 2 +1 \right)+(-5) = 4-5 = -1$$
Uh oh. At level 2 my maximum HP is 0. If I somehow manage to keep gaining levels, the situation becomes even more dire; the reduction of one maximum HP stays constant, so as I level up the most health I'm allowed to have goes deeper and deeper into the negatives.
Obviously this is a pretty contrived example, but it could come up in the future. As of now, I'm only aware of two sources of ability damage in 5e, and they hit INT and STR. But if a future book adds a way for your CON to be lowered, it could be disastrous. When your CON modifier is modified by 1, your maximum HP is affected by 1 point for each of your levels. The way the math works out, this means that if you want to calculate a character's maximum HP, you can assume that they've had the same CON modifier since level one; the result is identical. If I'm a level 20 Sorcerer or Wizard and my CON is somehow reduced to 1, my maximum HP immediately becomes -18!
…Is this really how it works? Or is there a rule saying (for example) that your maximum HP will always be at least one, preventing the issue?
Best Answer
As of the 10th printing, the minimum gained is 1 hit point.
The 10th printing (based on the errata published in November 2018) now indicates that the minimum gained at level up is 1 hit point. See this answer for more details.
The following is the original text of this answer, left as a historical footnote (or a way to justify the minimum of 1 prior to the errata).
There is no lower bound rule for hit points gained at level-up, but one of the designers seems to intend it.
Discussion on this lack of a rule has gone back several years. There are claims that a line appeared in a version of the D&D Next playtest (the precursor to 5th edition) that put a lower limit, but I can find no evidence of this publicly available. If there was such a rule, it did not make it into the Player's Handbook, and there has been no errata or Sage Advice to address the issue either. The by-the-book answer to your question is that there is no such rule.
It appears that one of the game's designers, Mike Mearls, believes there should be a 1 hit point minimum gained at each level-up (see this tweet in which he says the "minimum is 1"), but that is neither a rule nor an official ruling, since it was not included in the errata or in Sage Advice or stated by Jeremy Crawford, whose tweets can be considered authoritative.
However, if one of the rules designers believes such a rule is implicit, then it's probably safe to assume that's the way the game is intended to be played even if the statement was not from an (officially) authoritative source. If you are okay with an as-intended interpretation of the rules, it appears that you gain at least 1 hit point at each level-up.