At least 8 short rests could fit in a long rest, presuming your DM is on board with the shenanigans
With reference to the basic rules, a short rest is defined like so:
A short rest is a period of downtime, at least 1 hour long, during which a character does nothing more strenuous than eating, drinking, reading, and tending to wounds.
And a long rest is defined thusly:
A long rest is a period of extended downtime, at least 8 hours long, during which a character sleeps or performs light activity: reading, talking, eating, or standing watch for no more than 2 hours.
(Errata clarifies that you must sleep for at least six hours as part of a long rest, but that's not relevant to our problem - the overall durations are unaffected.)
Since a long rest requires a minimum of eight hours, and any given short rest requires a minimum of 1 hour, a character taking consecutive short rests while the rest of the party takes a long rest should be able to manage at least eight short rests in the period. If the rest of the party elects to continue the rest for a longer period, the character may be able to fit in another short rest for every hour their comrades continue lazing about. In most circumstances, though, one imagines that the adventuring party would not want to spend more time resting than they absolutely have to, so eight short rests would seem to be the standard.
Crawford Commentary (AKA Your DM Can Stop You)
Jeremy Crawford has weighed on this subject via twitter:
Is there anything stopping me from taking 8 consecutive rests during an 8 hour period?
The number of hours in a day is the cap on the number of short rests you can take that day. That said, a short rest is "at least 1 hour long" (PH, 186); notice the words "at least." If you spend 4 unbroken hours resting, for example, a DM could say that's one short rest.
After a short rest, the DM decides how much time must elapse or how much activity must occur before another short rest can start. Maybe 0 minutes, 1 minute, 10 minutes, or 1 hour. The key is that rests aren't meant to be a button you press. They're a narrative pause.
As he points out, the definition of a short rest requires a period of "at least 1 hour", and the intended way to play is that rests are defined narratively - so the DM is free to declare that a longer period of uninterrupted rest only constitutes a single short rest.
By cheesy RAW, the character can deliberately interrupt their current short rest after an hour passes simply by doing something strenuous, and then sitting down to rest again, so the period couldn't be defined as a single short rest. As a "strenuous activity" is not well-defined and could be no more than a few seconds of exercise, I assume the lost time devoted to forcing an end to your short rest is negligible when considering how many rests you can take and doesn't meaningfully change how many you can fit into a period.
However, Crawford goes on to state that the DM can decide how soon after finishing one rest the character must wait before they can start another, or how long the interrupting activity must actually be. If the DM is not on board with your shenanigans, this provides a Crawford-supported way to deny them - they can declare you just can't take consecutive short rests (we know the DM was always free to rule that way if they wanted, but now they have official backup, which may matter for some particularly argumentative players).
Actually, we're all elves
Sage Advice clarifies that, for elves:
If an elf meditates during a long rest (as described in the Trance trait), the elf finishes the rest after only 4 hours.
So in the special case where the whole party (or at least the rest of the party) are elves, the short-resting character only gets away with 4 rests while the rest of the party meditates.