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.
The 20th level is the highest you can achieve.
Dungeon Master's Guide p. 38
However you can still get more powerful. You may get an epic boon for every 30,000 XP you earn above 355,000 XP. (Dungeon Master's Guide, p. 231).
Epic boons offer a variety of things, including: immortality, truesight, nullifying damage, extra hit points, spell slots and much more.