What is the smallest resistance value of pull-up resistors for an I2C bus driving only PCF8575 chips?
I have a bunch of PCF8575 breakout boards (three to be exact) on one I2C network. Each with a 10K resistor, equivalent to a single 3.333k ohm resistor. I now appear to have one board that are no longer responding. Before I swap out the dead board out (again, already did once) I want to try to isolate the issue before I ruin a third board.
What is the smallest resistance value of pull-up resistors for an I2C bus driving only PCF8575 chips?
Depends on the bus voltage. As you didn't specify an Arduino model we expect an UNO so the bus voltage is 5V and the smallest resistor value is about 2k2 to stay below the maximum sink current of 3mA. Usually a 4k7 is a better choice for most setups.
Each with a 10K resistor, equivalent to a single 3.333k ohm resistor.
It's not recommended to have multiple pull-ups on one I2C bus. Additionally you have the internal pull-up of the UNO which shouldn't be relevant in this regard but you should definitely avoid multiple resistors and concentrate them in one pull-up per bus.
Did you calculate or measure the bus capacitance? Do you have long bus wires?