Word for "exploding error"?

Solution 1:

You could say the problems snowballed

to increase, accumulate, expand, or multiply at a rapidly accelerating rate

You might also invoke a comparison with the Hydra where

if you cut one hydra head, two more grow back

Other relevant terms might be "domino effect" and "cascading failure"

Solution 2:

Two other terms come to mind:

Mushroom or mushrooming (verb): Increase, spread, or develop rapidly (https://en.oxforddictionaries.com/definition/mushroom)

Where one problem or error triggers another, and that triggers another, and so on, they are often referred to as cascading errors or problems.

(https://en.oxforddictionaries.com/definition/cascade):

A succession of devices or stages in a process, each of which triggers or initiates the next.

also related definition of the term:

A large number or amount of something occurring at the same time

But cascading isn't generally used in the sense of multiplying exponentially.