This question was inspired by a scenario, where a character could use Elven trance and any number of short rests but no long rests. Based on this answer, I have a follow up question.
By the rules-as-written, is a player character required to engage in downtime activity?
Another way to ask the question is (per @SorcererQzot's comment) "Do characters actually have to stop adventuring between adventures?" since this question is based the discussions at the link.
Downtime Activities (p. 70, Basic Rules)
Between adventures, the DM might ask you what your character is doing
during his or her downtime. Periods of downtime can vary in duration,
but each downtime activity requires a certain number of days to
complete before you gain any benefit, and at least 8 hours of each day
must be spent on the downtime activity for the day to count. The days
do not need to be consecutive. If you have more than the minimum
amount of days to spend, you can keep doing the same thing for a
longer period of time, or switch to a new downtime activity. (The listed
pursuits are crafting, recuperating, practicing a profession,
researching, training).
There are a variety of things that a character can do during downtime, but is the actual taking of downtime required?
Taking a Short Rest is a period of downtime that seems to be an exception to the general case made above: if you want a Short Rest you have to be in a special "downtime" (at least 1 hour long per Basic Rules, p. 70) but it can occur in the middle of an adventure, and in the middle of the adventuring day.
Beyond that "in the field" downtime (which looks like an outlier) what is the hard requirement for downtime?
Answers must cite the rules as written, since the original question was raised in a pure RAW context.
Why is the Short Rest an outlier? Because "Adventurers can take short rests in the midst of an adventuring day" while the bulk of Downtime Activity happens between adventuring days. (p. 70 – 71, Basic Rules).
Best Answer
First off RAW gives the referee the authority to decide whether downtime activities are required.
In the PHB on page 187, we have a rule requiring the player to choose a lifestyle for their character and pay the appropriate expenses.
If your referee keeps strict track of time between adventure this expense can add up. If the income from adventuring doesn't cover it then the player may have to have their character do something in between adventures i.e. downtime activities to cover their expenses.
Now the lowest lifestyle is Wretched which cost nothing per day. However the description of Wretched is.
While there no direct mechanical consequence of living a Wretched lifestyle, the part highlighted in bold give the referee ample authority per RAW to make the character's life a hell between adventures. The character would win at first but likely be worn down by the incessant attacks and interruptions.
So while engaging in downtime activities is not specifically required, it would not be smart not to at least do something to insure a decent lifestyle..