What is the lethal dose ( LD50 ) of coffee?
According to one reference:

LD$_{50}$ for coffee = 118 servings at 240 ml each; and

LD$_{50}$ for espresso = 175 shots (volume not specified),

for a 75 kg person.

A much more comprehensive quantitative description is given here for the LD$_{50}$ of coffee for an 80 kg person (113 cups at 250 mL each, 135 mg caffeine per serving). They account for half-life of elimination of caffeine, and arrive a the conclusion that one can't actually reach the LD$_{50}$ (15,360 mg) for caffeine due to metabolism, elimination, and the limit on how much water can be processed by the kidneys.

Finally, as noted in the last reference and hinted at above, the amount of water a person would imbibe on the way to reaching the 113 cup total (about 30 L) would likely kill the person (water intoxication).