How much armor is required to get 100% less damage to physical attacks?

The formula is 100 / (100 + armor)

The percentage equivalent of the answer is the % of physical damage you'd take.

To reach 0% damage taken, or 100 % damage reduction, your armor value would have to be infinite

chances are you most likely couldnt get 100%

There's a formula somewhere on google. My maths are really poor, but basically you can never reach 100% due to the equation form.

There's diminishing returns on defenses.

The maximum amount of souls Thresh can collect is 999,999, granting him 749,999 bonus ability power and 749,999 bonus armor (99.99% damage reduction, reduced to 99.98% when hit by Last Whisper)

From the wikia.

For all intents and purposes thresh is the highest at 99.99% with his passive as said before and amumu with his E passive and a lot of armor is capable of taking 0 physical damage from minions (if were not counting baron empowered/super)