(n.) the deliberate use of a loaded term or overstatement for rhetorical effect
When a minor inconvenience is labelled a disaster, that’s an example of auxesis.
There are lots of terms for exaggeration in rhetoric, the most familiar of which is probably hyperbole. In general, auxesis is just another name for hyperbole: both words describe a statement in which something is described in often ludicrously disproportionate terms, for rhetorical or humorous effect.
But while hyperbole (which essentially means ‘thrown beyond’) tends to be used of an individual instance in which something is described in overblown terms (“I’ve told him a million times!”), auxesis is often used more specifically when a subtle or neutral word is substituted for a more loaded or dramatic one. So while “I’ve told him a million times!” is hyperbole, following that up with “He refuses to listen!” is auxesis—because implying that someone is actively refusing to take advice is different to saying that they simply do not listen. Likewise, people who go to pieces over the slightest inconvenience are likely guilty of auxesis; the term applies to minor events being blown into catastrophes too.
That being said, auxesis is also used in a much less specific way too, to describe rhetorical exaggeration in all its forms. Auxesis essentially means ‘increase’ or ‘amplification’ in Greek, and so is sometimes called upon as an umbrella term to describe all rhetorical tropes that somehow amplify what is being said.
It’s a confusing state of affairs then, as auxesis is used both as a synonym for hyperbole and as a more specific form of hyperbole—but also as a general term for all related forms of exaggeration, including hyperbole itself.