Typically a cadence of one resupply mission every couple months is maintained. If this cadence were to be interrupted for whatever reason, how long could, say, the current crew size of three members last onboard before needing to abandon ship due to lack of supplies? Have we ever come close to this limit?
ISS operations plans re-boosts to have roughly 90 days before the vehicle gets too low. This doesn't mean that re-supply missions are 90 days apart since the ISS is often boosted much higher than this minimum level -- especially post-Shuttle. But, it gives you a rough idea of the timeframes involved.
I suspect that other consumable margins are larger than this since there is always a Soyuz seat for each crew member. This is especially true if you reduce the crew size to extend the supplies.
I looked into this a while ago. It's easy to get bogged down in the details, because although the data is very public, there are so many parameters for every mission that I found it hard to get a straight answer. I used these points to "dumb down" the question:
- Seeking maximum time between launch of one mission and the next mission
- Manned and unmanned missions are counted
- Content of the mission is ignored
Someone involved might hem and haw due to a number of inadequacies with this. For example, the payload doesn't reach the station on the same orbit (and maybe not even the same day) of the launch. The Wikipedia pages on the list of space flights to the ISS is also quite concerned with the length of time that every spacecraft remains docked to the ISS. There is some valid contention about how to handle return trips to Earth, because over some interval it may have received nothing from Earth but still sent something to Earth. I'm not interested in return trips because it doesn't supply the station with new resources, nor do I care about the time taken to reach the station because all I care about is how long people survive in orbit with resources already in orbit. I find all of these details highly tangential to the straightforward query of how long have we had people up there without sending anything more up there.
Now, the record holders:
- Nov 25, 2002 to April 2, 2003 -> for a total of 128 days
- April 28, 2003 to Oct 27, 2003 -> for a total of 125 days
As I understand it, these extended intervals were connected to the Space Shuttle Columbia disaster, which happened on Feb 1, 2003. Now I could have potentially had a data processing error, but in a simplistic sense, these seem consistent with what you can read about those missions.
The story is that the ISS was, indeed, sort of struggling during those time frames. Because of that, these records should be within striking distance of the maximum time they could go without resupply. However, there are other complications. I would hope that since 2003 they've better prepared for a similar scenario. Plus, there's the simple fact that as modules are added, the mass to area ratio increases which would give more time for station keeping. The ISS also has experimented with plasma/ion thrusters, and this could keep it in orbit for longer. Supplies are an issue, but I don't think there's any fundamental limit to how much could be stored. If the time frame for those things was extended, they could probably push the time they can go without resupply to close to the human records for time in microgravity, but I don't think there's any reasons for them to spend resources on this objective just for the sake of a record.
Mir went for longer times without resupply, and I find a maximum of 240 days, which is almost twice what the ISS record was. They had a smaller crew and experimented with keeping them up there for longer periods of time.