Negative Racial Ability Modifiers
There are a couple of racial choices to actually get below a three in character creation.
In Volo's Guide to Monsters you will find the Kobold and Orc racial options which give the character -2 negative ability modifiers to strength and intelligence respectively. This is an actual means by which a player character can, with rolled stats, begin with less than a three. Thus those who have their heart set on roleplaying an orc barbarian with an intelligence of 1 can take heart, the dream is possible even if it would take some very "lucky" rolls.
This is by no means the most important time when sub-three scores would come up. Most players will probably only ever encounter it for certain non-player creatures like those Blckknght's answer lists, which they may run as a DM, wildshape into, have as familiars, etc. It may also come up for them when suffering from temporary negative ability effects, though this would typically only get them below three if they started with very low stats. However, given the context of the modifier chart being, as you noticed, adjacent to character stat generation in the PHB, it may well be that WotC also anticipated that they might eventually add player races with negative stat bonuses, which had existed in most prior editions of the game, though they have clearly made a decision to mostly avoid them in 5e.